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AFWL {WLIL-2)
KIRTLAND AFB, H MEX
HIGH FREQUENCY
PROPERTIES OF PLASMA
K. D. SinePnikov, Editor-in-Chief
Academy of Sciences, Ukrainian SSR,
Izdatel'stvo "Naukova Gumka,"
Kiev, 1965
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION • WASHINGTON, D. C. # MARCH 1967
TECH LIBRARY KAFB, NM
QQb^D3M
NASA TT F-449
HIGH FREQUENCY PROPERTIES OF PLASMA
K. D. Sinernikov, Editor-in-Chief
Translation of "Vysokochastotnyye Svoystva Plazmy.
Academy of Sciences, Ukrainian SSR,
Izdatel'stvo "Naukova Dumka,"
Kiev, 1965.
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
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TABLE OF CONTENTS
HIGH FREQUENCY PROPERTIES OF PLASMA
Page
SECTION I: HIGH jFREQU ENCY PLASMA HEATING 1
INVESTIGATION OF THE ENERGY OF CHARGED PARTICLES EMANATING
FROM A MAGNETIC TRAP DURING HIGH FREQUENCY HEATING
N. I. Nazarov, A. I. Yermakov, V. T. Tolok 1
MEASUREMENT OF THE PERPENDICULAR ENERGY COMPONENT AND THE
PLASMA DECAY TIME DURING HIGH FREQUENCY HEATING
N. I. Nazarov, A. I. Yermakov, V. T. Tolok 6
HIGH FREQUENCY ENERGY ABSORPTION BY A PLASMA IN ION
CYCLOTRON RESONANCE IN STRONG, HIGH FREQUENCY FIELDS
V. V. Chechkin, M. P. Vasil'y^v, L. I. Grigor'yeva,
B . I . Smerdov 11
INVESTIGATION OF CONDITIONS PRODUCING A DENSE PLASMA IN
A METALLIC CHAMBER AND ITS HIGH FREQUENCY HEATING
0. M. Shvets, S. S. Ovchinnikov, V. P. Tarasenko,
L. V. Brzhechko, 0. S. Pavlichenko, V. T. Tolok 22
HIGH FREQUENCY PLASMA HEATING
K. N, Stepanov 33
DIELECTRIC CONSTANT OF A PLASMA IN A DIRECT PINCH MAGNETIC
FIELD AND IN A DIRECT HELICAL MAGNETIC FIELD
V. F. Aleksin, V. I. Yashin 55
SECTION II: LINEAR PLASMA OSCILLATIONS 64
KINETIC THEORY OF ELECTROMAGNETIC WAVES IN A CONFINED PLASMA
A. N. Kondratenko 64
iil
Page
KINETIC THEORY OF A SURFACE WAVE IN A PLASMA WAVE GUIDE
M. F. Gorbatenko, V. I. Kurllko 75
SINGULARITIES OF AN ELECTROMAGNETIC FIELD IN A NONUNIFORM,
MAGNETOACTIVE PLASMA
V. V. Dolgopolov 80
EXCITATION OP A MAGNETOHYDRODYNAMIC WAVE GUIDE IN A
COAXIAL LINE
S. S. Kalmykova, V. I. Kurllko 89
THEORY OF MAGNETOHYDRODYNAMIC WAVE SCATTERING AT THE END
OF A WAVE GUIDE
V. I. Kurllko 91
DETERMINATION OF PLASMA TEMPERATURE AND DENSITY DISTRIBUTION
BY REFRACTION AND DAMPING OF A BEAM
V. L. Sizonenko, K. N. Stepanov 100
SECTION III: PLASMA NONLINEAR OSCILLATIONS AND WAVE INTERACTION 110
EXCHANGE OF ENERGY BETWEEN HIGH FREQUENCY AND LOW FREQUENCY
OSCILLATIONS IN A PLASMA
V. D. Fedorchenko , V. I. Muratov, B. N. Rutkevich 110
DISSIPATION OF PLASMA OSCILLATIONS EXCITED IN A CURRENT-
CARRYING PLASMA
Ye. A. Sukhomlln, V. A. Suprunenko, N. I. Reva,
V. T. Tolok 119
DAMPING OF INITIAL PERTURBATION AND STEADY FLUCTUATIONS
IN A COLLISIONLESS PLASMA
A. I. Akhlyezer, I. A. Akhiyezer, R. V. Polovln 127
CHARGED PARTICLE INTERACTION WITH A TURBULENT PLASMA
I . A. Akhlyezer 133
IV
Page
THEORY OF NONLINEAR MOTIONS OF A NONEQUILIBRIUM PLASMA
I . A. Akhlyezer 136
NONLINEAR PROCESSES IN A UNIFORM AND ONE-COMPONENT PLASMA
N. A. Khizhnyak, A. M. Korsunskiy 143
INDUCED SCATTERING OF LANGMUIR OSCILLATIONS IN A PLASMA
LOCATED IN A STRONG MAGNETIC FIELD
V. D. Shapiro, V. I, Shevchenko 151
NONLINEAR THEORY OF LOW FREQUENCY OSCILLATIONS EXCITED
BY AN ION BUNDLE IN A PLASMA
D . G. Lomlnadze , V. I . Shevchenko 159
NONLINEAR PHENOMENA IN A PLASMA WAVE GUIDE (ION
CYCLOTRON RESONANCE AT A DIFFERENCE FREQUENCY)
B. I. Ivanov 172
SECTION IV: EXCITATION OF PLASMA OSCILLATIONS 180
RADIATION OF ELECTRONS IN THE PLASMA-MAGNETIC FIELD
BOUNDARY LAYER
V. V. Dolgopolov, V. I. Pakhomov, K. N. Stepanov 180
RADIATION OF LOW FREQUENCY WAVES BY IONS AND ELECTRONS
OF A NON-ISOTHEEMIC MAGNETOACTIVE PLASMA
V. I. Pakhomov 183
BRAKING OF RELATIVISTIC PARTICLES IN LOW ATMOSPHERIC
LAYERS
V. B. Krasovltskiy , V. I. Kurilko 199
EXCITATION OF WAVES IN A CONFINED PLASMA BY MODULATED
CURRENTS
A. N. Kondratenko 203
HIGH FREQUENCY PROPERTIES OF PLASMA
PLASMA PHYSICS AND THE PROBLEMS OF CONTROLLED THERMONUCLEAR REACTION
ABSTRACT
These articles present the results derived from
theoretical and experimental investigations of high
frequency plasma properties : the methods of high fre-
quency plasma heating, propagation of electromagnetic
waves in a magnetoactive plasma, thermal radiation of
a plasma, and development of instabilities when employing
high frequency methods of plasma heating. A description
is provided of experimental equipment developed for high
frequency heating and containment of a plasma.
This collection is designed for scientific researchers
and engineers dealing with the problems of a plasma and
its technical application, as well as for students and
graduate students in the physics departments of universi-
ties and physical-technical institutes.
SECTION I
HIGH FREQUENCY PLASMA HEATING
INVESTIGATION OF THE ENERGY OF CHARGED PARTICLES EMANATING
FROM A MAGNETIC TRAP DURING HIGH FREQUENCY HEATING
N. I. Nazarov, A. I. Yermakov, V. T. Tolok
The methods of resonance excitation of the eigen fluctuations of a /5*
plasma cylinder by outer electromagnetic fields have been extensively em-
ployed in plasma heating with high frequency fields. Spatially periodic
electromagnetic fields may be employed to excite the eigen fluctuations in
a plasma located in a constant magnetic field to frequencies which are close
* Note: Nimibers in the margin indicate pagination in the original foreign
text.
to the gyrofrequency of ions co « co . (an ion cyclotron wave) or to frequencies
ojjj^ < 0) « (jOjjg (rapid magnetosound wave) , where m^^ is the electron gyrofre-
quency and tOti- is the ion gyrofrequency. In the first case, the energy of
electromagnetic fluctuations is transmitted directly to the ions, and in the
second case the energy is transmitted to the plasma electrons.
The articles (Ref. 1, 2) have investigated the conditions for the reso-
nance excitation of these fluctuations, their propagation, and damping. It
has been found that even at high electron temperatures the damping is signi-
ficant and is caused by a collisionless mechanism. It was also found that, /6
when this excitation method is employed, high frequency power is transmitted
to the plasma very effectively.
This study presents the results derived from measuring the energy of
ions and electrons passing along the magnetic field, when ion cyclotron and
rapid magnet ohydrodynamic waves are employed for heating the plasma. The
experiment was performed on a "Sneg" apparatus, which has been described in
great detail previously (Ref. 1). The eigen fluctuations were excited in the
plasma by spatially periodic electromagnetic fields at a frequency of 10 Mc
with the appropriate selection of the magnetic field strength Hg . In contrast
to preceding experiments, the power of the high frequency (hf) generator was
increased to 300 kw.
In order to increase the power introduced into the plasma, the pulse of
the hf generator was programmed so that the strong loading on the
circuit at the moment of its resonance loading by the plasma was compensated
by a corresponding voltage increase in the pulse from the hf generator (Figure
1). Thus, the necessity of a special electrical strengthening of the hf cir-
cuit was avoided, even when a power greater than 100 kw was introduced into
the plasma.
The energy of the charged particles was measured by a transit time electro-
static analyzer (Ref. 3) and a multigrid probe (Ref. 4). The first method
made it possible to study the energy spectrum and the mass composition of
plasma ions; the second method made it possible to measure the energy of
ions and electrons. The plasma electron temperature was determined by a
spectral method.
The input slit of the analyzer, which was located 25 cm behind a mag-
netic mirror, cut out a narrow plasma flux, from which an ion bundle was
separated after passing a separation device. The energy of the bundle
was analyzed by the electric field of the flat condenser. Ion fluxes were
recorded by an ion-electron converter, which changed the ion bundle into a
bundle of electrons accelerated up to 20 kev, after deflection in the
analyzing condenser. These electrons were detected by a plastic scintilla-
tor with a photoelectric multiplier.
In order to study the mass composition of the plasma ions, the flight
Figure 1
time was measured by ions having a drift section 56 cm long. When the analy-
zer modulator was supplied with voltage having a rectangular form, it was
possible to obtain short pulses of the ion current (t = 1; 0.5; 0.2 microseconds)
Due to a difference in the velocities of different ions, the ions were separ-
ated by mass in the drift space. The flight time of ions in the drift sec- /7
tion was used to determine the ion velocities and masses, respectively. By
measuring the amplitude of the current signals at different periods of time
and by changing the voltage on the analyzing condenser, it was possible to
record the energy spectra of ions having different masses and to observe
the change in ion energy during heating, by means of this analyzer.
In order to exclude the scatter of ion current pulses, the result was
averaged over ten measurements (the ion energy was determined with the
analyzer with an accuracy of 8%) .
Figure 2 presents the oscillograms of a typical signal from the photo-
multiplier, whose magnitude was proportional to the current of ions having
an energy of 1500 ev — a; b — represents the suppression signal of micro-
waves having a wavelength of 8 mm.
Figure 3 shows the energy spectra of plasma protons (the coordinates:
the distribution functions i|; — ion energy E.) when the plasma is heated by
means of ion cyclotron waves for two voltages on the hf circuit (curve 1 —
U(, = 28 kv; curve 2 — U = 32 kv) . The plasma density to be measured during
heating was no less than 2*10^^ cm~^. As may be seen from the figure, the
energy at the spectrvim maximum amounts to 2 kev. With an increase in the
voltage Ug on the exciting coil of the hf circuit, the mean ion energy in
the spectrum increases proportionally to the square of the voltage U^. How-
ever, during the heating pulse it remains almost constant, which points to /8
large losses which are apparently caused by overcharging.
For purposes of comparison, the proton energy spectrum is presented which
was recorded when the plasma was heated by a rapid magnetosound wave with a
' ^-^,^
Figure 2
voltage of 32 kv on the hf circuit (Figure 4)
trum maximum is 150 ev In all in this case.
The ion energy at the spec-
As may be seen from the ion mass-spectrograms (Figure 5) when the plasma
is heated by an ion cyclotron wave, there are only hydrogen ions HJ, H|', EJ
in the plasma. The presence of these ions is apparently due to the fact that
the plasma electron temperature is low (20-25 ev) . It is particularly inter-
esting to note that all three types of hydrogen ions have approximately the /9
same energy, although the resonance acceleration occurs only for H^.
For purposes of control, the ion energy was measured by another method
— multigrid probe with retarding potential (Ref. 4). This made it possible
to record the energy spectrum of electrons. The multigrid probe was placed
at a distance of 10 cm from the magnetic mirror. The plasma density was
greatly reduced by means of a diaphragm having several 0.1 mm openings. It
was possible to separate an electron bundle or an ion bundle, depending on
the sign of the pulling voltage on the first grid; the bundles were analyzed
by the retarding potential which was supplied to the second grid. In order
to decrease the possibility of ionization, a differential pumping was employed
to maintain a vacuum of 6.7-10-'* n/m^ within the probe walls.
'f'.
rel.
unxt
1
15
/
AJ
2
\
10
//
A
5
n
/
y
500 1000 tSOO 2000 2500 Ei.ev
Figure 3
100 200 300 £,js.v
Figure 4
Figure 5
The results derived from measuring the Ion energy spectrtjm by means of
the multi-grid probe confirmed the results obtained with analyzer measure-
ments. The proton energy spectrum maximum was in the region of 2 kev when
the plasma was heated with Ion cyclotron waves. The electron energy remained
low, and amounted to 30 ev in all. This result also coincides closely with
the results derived from measuring the electron energy by the spectral
methods.
As would be expected, the measurements of the energy spectrvnn of ions
and electrons by the multigrid probe, when the plasma was heated with a
rapid magnetosound wave, showed that the electron and ion energies were
approximately the same (150 ev) . The ion energies obtained by measurements
with the probe and the analyzer also coincided fairly well.
REFERENCES
Nazarov, N. I., Yermakov, A. I., Lobko, A. S. , Bondarev, V. A., Tolok,
V. T., Sinel'nikov, K. D. Zhurnal Tekhnicheskoy Fiziki, 32, 5, 536,
1962.
2. Nazarov, N. I., Yermakov, A. I., Dolgopolov, V. V., Stepanov, K. N. ,
Tolok, V. T. Yadernyy Sintez, 3, 255, 1963.
3. Kalmykov, A. A., Timofeyev, A. D. , Pankrat'yev, Yu. I., Tereshin, V. I.,
Vereshchagin, V. L. , Zlatopol'skiy, L. A. Pribory i Tekhnlka Ek-
sperlmenta, 5, 142, 1963.
4. Bulyginskly, D. G., Galaktinov, B. V., Dalmatov, K. A., Ovsyanikov, V.A.
Zhurnal Tekhnicheskoy Fiziki, 33, 183, 1963.
MEASUREMENT OF THE PERPENDICULAR ENERGY COMPONENT AND THE /lO
PLASMA DECAY TIME DURING HIGH FREQUENCY HEATING
N. I. Nazarov, A. I. Yermakov, V. T. Tolok
The correct determination of the charged particle temperature is of
paramount importance when studying the effectiveness of plasma heating.
One convenient method of determining the plasma temperature consists of
measuring its thermal diamagnetism by an external diamagnetic probe which
includes a coliamn of the plasma to be heated. The method is based on
measuring the difference between the strengths of the magnetic field out-
side and within the plasma AH, which is a function of the gasokinetic
(-^)
pressure. For a plasma with small g j g = ^^ j , this difference is deter-
mined by the expression
AH = Ho - H^ithin =
= itIE (1)
Ho
where p is the gasokinetic pressure. With a plasma having quasineutrality,
we have
p = nk {Tj_i + T^,),
(2)
where n is the plasma density; k is the Boltzmann constant. T,. and Tj^
are the perpendicular ion and electron temperatures, respectively. Thus,
by measuring AH and knowing the plasma density, we may compute its tempera-
ture.
The temperature was measured by this method on a "Sneg" apparatus
Figure 1
(Ref. 1). Similar measurements were performed in (Ref. 2). The plasma
was either produced by a powerful ion cyclotron wave (Ref. 1), or by a
rapid magnetosound wave (Ref. 3). Hydrogen in the 0.13-0.4 n/m^ pressure
range was used as the process gas. The waves were excited at a frequency
of fg = 10^ cps, and the pulse high frequency power, transmitted to the
plasma, was 100 kw. In order to avoid transitional processes related to
the sharp change in plasma density during the initial period, a coupled,
high frequency pulse, whose envelope is shown in Figure 1, was supplied to
the exciting coil. The first pulse produced a plasma, and the second pulse
was employed to heat it. The duration of the pulses, the amplitude, and
the interval between them could be changed Independently over very wide
Intervals.
The period during which the strength of the pulse magnetic field Hg
changed was 24 microseconds. The strength of the field was selected so
that either the ion cyclotron wave, or the rapid magnetosound wave, was
excited resonantly at the time the second high frequency pulse came into
operation. The plasma density was measured by a microwave Interferometer
at the wavelengths 8.2 and 4 mm.
/ll
The quantity AH, which was caused by the plasma dlamagnetism, was de-
termined by measurements of the electromotive force (emf) by a dlamagnetlc
probe. The probe consisted of two colls, one of which Included the plasma
column. The other coll was employed to compensate for the emf caused by a
change in the strength of the confining magnetic field Kg. In order to
eliminate the emf produced by the propagated ion cyclotron wave, a five-unit
low-frequency filter was employed, which Intersected all frequencies above
3 Mc. The probe was located in the region of the "magnetic beach", at a
distance of 30 cm from the edge of the exciting coil. In order to decrease
the effect of attenuation of the dlamagnetlc signal, due to reflection of
the charged particles from the walls, the diameter of the discharge chamber
was increased up to 8 cm in the region of the dlamagnetlc probe, while the
Figure 2
diameter of the plasma with n > 10-^^ cm~^ equalled 3.5 cm-
Figure 2 shows an oscillogram of the dlamagnetlc signal — a, obtained
when the plasma was heated with an ion cyclotron wave; b — represents the
Interferogram of an 8-mm signal. The gasokinetlc plasma pressure increased
very rapidly (in~10 microseconds), and then barely changed until the end
of the high frequency pulse. Since the plasma density changed very little /12
— (1.2 - 1.5) X 10^^ cm~^ — during the high frequency pulse, and since
it decreased slowly after it had ended (t ~ 270 microseconds) , it can be
assumed that the dlamagnetlc signal was proportional to the rate at which
the perpendicular energy component of the plasma -r— (Ti , + Tj^) changed.
Figure 3, a shows an oscillogram of a dlamagnetlc signal. In order to deter-
mine the plasma temperature, AH was calculated after integration of the dla-
magnetlc signal. The integrated dlamagnetlc signal is shown in Figure 3, b.
The total value of T, thus obtained for resonance excitation of an ion cyclo-
tron wave, amounted to 1 kev. The electron temperature, determined by the
spectral method, was in this case 20-30 ev. Thus, the ion temperature was
measured indirectly. The plasma temperature was determined by this same
method when it was heated by a rapid magnetosound wave. In this case
T = 200-300 ev. The values of T, obtained according to the dlamagnetlc
signal, closely coincide with the measured energy of charged particles em-
Inatlng from the system along the magnetic field (Ref. 4). The small dl- /13
vergence between T. and T,, is caused by the fact that measurements of the
plasma temperature according to the dlamagnetlc signal give an average (over
the plasma column cross section) temperature.
Ho
Figure 4 shows the dependence of the ion temperature T-j^ on (Hq —
"ci
the strength of the outer magnetic field, H . — the strength of the magnetic
field at which the gyrof requency of a proton equals 10 Mc) . It can be seen
that maximum heating occurs with resonance excitation of an ion cyclotron
wave.
Figure 3
'..-
kev
IP
f \
(\7^
' \
as
0^5
1
V
J
\
^
-.
.
I
V 12
Figure 4
The ion temperature barely changes during heating (see Figure 3) .
There is a very rapid temperature decrease after the high frequency pulse
is recorded. This change in the ion temperature points to the presence
of great losses. Since the ion temperature is considerably greater than
the electron temperature when this method of plasma heating is employed,
it may be assumed that one of the mechanisms for rapid ion cooling is their
energy loss when colliding with electrons. However, for a plasma with
Hg ~ 2*10^^ and Tg~ 25 ev, the time required to cool hot ions must be 100
microseconds. In actuality, the cooling time equalled 10 microseconds.
Therefore, there is no basis for assuming that this is the main loss
mechanism.
Another mechanism for rapid ion cooling may be their overcharging,
since under the experimental conditions the highly-ionized plasma column
was surrounded with a weakly-ionized cold plasma which was in contact with
the discharge chamber walls. For a plasma with an electron density on the
^T3
order of 10^^ cm-^ and a neutral gas pressure at the chamber walls of
approximately 1.33 "10"^ n/m^, the probability of overcharging exceeds the
ionization probability. Consequently, the losses to overcharging may be
considerable, and they continue to increase with an increase in the ion
IS
fO
• . ' '
micros ec
*%
N
1
5
\
m 200 600 iOOO ZOOOfi^V
Figure 5
energy. Figure 5 shows the dependence of the ion cooling time on their
energy. The solid line corresponds to the time required for overcharging
the ions for a density of the surrounding gas of ng = 6* 10-^^ cm~^. This
density was selected so that a comparison could be made between the value
obtained theoretically and the experimentally measured value t-j, for
T-j^ = 200 ev. The nature of the dependence of T-p. on T- shows that under
experimental conditions the energy losses are primarily caused by ion
overcharging .
Thus, these experiments enable us to draw the conclusion that a /14
plasma may be heated to a temperature exceeding 1 kev by means of resonance
excitation of an ion cyclotron wave. The limiting value T^ is determined
by the apparatus parameters. In addition, the results obtained provide a
basis for assuming that — when a hot plasma is insulated from the chamber
walls by "vacuum interstratif ication" — the time the plasma may be contained
may be increased considerably, under the condition that there are no other
loss mechanisms.
REFERENCES
Nazarov, N. I., Yermakov, A. I., Lobko, A. S. , Bondarev, V. A., Tolok,
V. T., Sinel'nikov, K. D. Zhurnal Tekhnicheskoy Fiziki, 32, 5, 536,
1962.
Hooke, W. H., Rothman, M. A., Adam, I. Bull. Am. Phys. Soc. Ser. II, 8,
174, 1963.
Nazarov, N. I., Yermakov, A. I., Dolgopolov, V. V., Stepanov, K. N. ,
Tolok, V. T. Yadernyy Sintez, 3, 255, 1963.
10
4. Nazarov, N. I., Yermakov, A. I., Tolok, V. T. Present Collection, 5.
HIGH FREQUENCY ENERGY ABSORPTION BY A PLASMA IN ION CYCLOTRON /15
RESONANCE IN STRONG, HIGH FREQUENCY FIELDS
V. V. Chechkin, M. P. Vasil'yev, L. I. Grlgor'yeva,
B . I . Smerdov
This article represents a continuation of studies we performed pre-
viously (Ref . 1, 2) on high frequency energy absorption by a plasma in
ion cyclotron resonance.
As is well known (Ref. 3-5), the heating of a plasma by a variable
field at a frequency which is close to the ion cyclotron frequency is very
effective, if there are mechanisms leading to energy thermalization of the
orderly motion of plasma particles in the field of an ion cyclotron wave.
High frequency energy absorption by the plasma may occur, in particular,
due to "close" collisions of ions which are in resonance with other types
of ions, electrons, and neutral atoms in a cold plasma. This absorption
may also be due to "collisionless" cyclotron damping which is caused by the
thermal motion of ions in a high temperature plasma. In both cases, it is
assumed that the amplitude of the high frequency field is fairly small, so
that ions receiving energy in the wave field can transmit it to other parti-
cles. If this condition is not fulfilled, nonlinear processes must arise
in the plasma, due to which the ion distribution function is essentially
distorted. Distortion of the ion distribution function by the ion cyclotron
wave with a finite amplitude leads to a decrease in the wave absorption co-
efficient down to a small value which equals, in order of magnitude, the
absorption coefficient for pair collisions at a given temperature.
The expression obtained in the case of a high-temperature plasma
(Ref. 5) for a critical field strength of an ion cyclotron wave — which
leads to a significant distortion of the ion distribution function when
it is exceeded — can be written as follows
((ot)
2Tr
where w is the wave frequency; k . = r — ; A — axial wave length; H — con-
II A
stant magnetic field; t — relaxation time of ions due to ion- ion colli- /16
sions; N. — ion density; T^ — ion temperature.
11
In the case of a cold plasma, the critical strength of the wave
electric field (at the absorption maximum) is as follows
Ecr--^ l/i;. (2)
^^eff V M
When this value is reached, the velocity of the ordered motion of an ion
liquid, with respect to electrons, equals the thermal ion velocity. M —
ion mass; e — ion charge; Yeff — effective frequency of ion collisions.
If the velocity of the relative motion of ions and electrons in the
field of an ion cyclotron wave is greater than the ion thermal velocity,
"bunched" instability may arise in the plasma, which is related to the excita-
eH
tion of high frequency (as compared with 'J^i=T7~) longitudinal fluctuations,
vifcose increasing increment is considerably greater than the cyclotron ion frequency,
and the wavelength is considerably less than the ion cyclotron wavelength.
The excitation of these fluctuations by an ion bundle moving in a direction
which is perpendicular to a constant magnetic field has been investigated in
(Ref. 6-8).
Similar small-scale electrostatic plasma fluctuations, which take place
in a wide frequency and wave number range, must lead to an increased exchange
of energy between plasma ions and electrons (as compared with the exchange
caused only by Coulomb collisions), and must also lead to a significant in-
crease in all the transfer coefficients across a constant magnetic field.
Let us examine certain results of experimental studies in this light
(Ref. 9, 10). In these studies, the field strength of an ion cyclotron
wave exceeds by at least one order of magnitude the critical field strength
(1) , and the time of ion-ion relaxation is either a comparable with the
plasma decay time (Ref. 9) or considerably exceeds it (Ref. 10). For this
reason, the strong high frequency energy absorption by the plasma and the
ion heating, observed in these studies, cannot — in our opinion — be
caused by cyclotron damping. In addition, the study (Ref. 11), which was
carried out on the same apparatus as was employed in (Ref. 10), observed a
rapid plasma decay if the high frequency power introduced into the plasma
exceeded a certain critical value. This rapid decay was apparently caused /17
by "bunched" instability.
The phenomenon of anomalous plasma diffusion across a magnetic field,
produced for a critical value of a high frequency field strength with a
frequency close to ion cyclotron frequency, was discovered and studied in
detail in (Ref. 2). In particular, this study found that increased diffu-
sion occurs if the velocity, acquired by ions in an azimuthal high frequency
field during the period between collisions, is comparable to the thermal ion
velocity or exceeds it — i.e., if relationship (2) is fulfilled.
The study (Ref. 12) performed an experimental determination of the
12
increase in the effective frequency of ion collisions in a low-density
plasma under conditions of ion cyclotron resonance in strong, electric,
variable fields. It was found that the dependence obtained experimentally
for the frequency of ion collisions on the high frequency field strength
cannot be explained on the basis of a theory postulating ion collisions
with neutrals. The assumption was advanced that such a dependence is
caused by 'bunched" instability which arises in strong variable fields.
We can clarify the nature of this absorption by comparing the data
obtained experimentally, regarding high frequency energy absorption by
a cold plasma close to ion cyclotron resonance, with the theory advanced
in (Ref. 5). Let us establish a relationship between the effective fre-
quency of ion collisions and anomalous plasma diffusion in ion cyclotron
resonance, which was studied in (Ref. 2). Finally, by making certain
numerical estimates, we can show that both high frequency energy absorp-
tion by the plasma, and anomalous plasma diffusion in fields with super-
critical strength values, may be caused by "bunched' instability produced
in the field of an ion cyclotron wave (this instability was investigated
theoretically in [Ref. 8]).
Description of the Apparatus .
Measurement Methods
The apparatus which was employed for the study was described in de-
tail in (Ref.l). The plasma was produced by pulse discharge with oscil-
lating electrons in hydrogen, in a 7'10~^— 7 n/m^ pressure range. The
diameter of the glass discharge tube was 6 cm, and the distance between /18
the cathodes was 80 cm. The strength of the longitudinal, quasi-constant
magnetic field could be changed between 4*10'* — 6.4-10^ a/m.
High frequency energy was introduced into the plasma by means of
an artificial LC-line which was connected to the self -excited oscillator with
a frequency of 7. 45 -10^ cps. During excitation at this frequency, 2.5 wave-
length oscillation was applied to the section of the line x^hich was slipped on
to the discharge tube. This corresponded to an axial high frequency field
period of 23 cm. By changing the anode strength of the oscillator, and also
the connection between the line and the oscillator, it was possible to change
the amplitude of the high frequency azimuthal line current within 0.5 - 35
a/ cm.
The oscillator was switched on for approximately 100 microseconds
after the pulse of the discharge current had terminated. As is shown in
(Ref. 1), the maximum high frequency power which could be absorbed by a
plasma in resonance was 18 kw for a plasma density of 1.7'10'^^ cm"^, and
an azimuthal line current of 30 a/cm.
This article presents the measurements of plasma density, electron
temperature, and the high frequency power absorbed by the plasma. The
13
■^
*
0.3
f
'
I
/
0.2
0,1
/
y
/
/
/ 2 3 N-10,cn
Figure 1
13
electron plasma density and its change with time in the (1.7 - 0.25) 10
range was measured by means of an interferometer at a wavelength of
cm
-3
8.1 mm = 0.81 cm. The experimental point corresponding to a density of
„-3
3*10 ^ cm was obtained by extrapolation of the plasma density dependence
on time toward large densities (Figure 1) .
The plasma electron temperature was determined according to the de-
pendence of the luminosity intensity of the line Hg on the high fre-
quency LC-line current [see (Ref. 2)]. The luminosity intensity of the
line Hg was measured by means of a UM-2 monochromator and a photoelectron
multiplier. Due to anomalous diffusion, the plasma decay time, for line
currents on the order of 5 a/cm and above, was comparable with the electron
lifetime between two collisions leading to excitation of a neutral atom. /19
For this reason, for line currents which were greater than, or approxi-
mately equal to, 5 a/cm, the electron temperature which was computed
according to the line intensity Hg could be too low.
The high frequency power absorbed by the plasma was measured with an
all purpose meter of transmitted power which was described in (Ref. 13).
The recorders for the current and the strength were connected to the line
at the point where it was attached to the solenoid. By means of two inter-
changeable current recorders with different sensitivity, it was possible to
measure the power absorbed by the plasma, corresponding to a line current of
0.5-10 a/cm.
All of the measurements described in this article were performed at
a hydrogen pressure of 0.133 n/m^.
14
Nature of High Frequency Energy Absorption
Under the conditions of our experiment, the high frequency power
absorbed per unit of plasma cylinder length is (Ref. 5)
^ = S ^' (* • ^) (?)'«'. i"J W s5n^& ,z, (3)
where j q is the amplitude of the azimuthal high frequency current in the
coil per unit of cylinder length; R — coil radius; a — plasma radius
(it is assumed to equal the inner radius of the discharge tube in all of
the computations); Ki — McDonald function; kn = T — — axial wave number
(it is determined as the axial period X of high frequency current in the
coil); 0) — oscillator frequency; nn = ; f(X) — the function whose
specific form depends on the nature of the high frequency energy absorp-
tion. In particular, for collision absorption we have
/ {X) = lef f_i (4)
where 2
Strictly speaking, relationship (3) is only valid in the case of /2Q
long-wave fluctuations (or a small plasma filament radius), when k || a«l
and kj_ a << 1 (k_|_ — radial wave number). Under our conditions, both of
these quantities are on the order of unity. However, as the computation
showed, the error produced when equation (3) is applied to our case is
small .
Under the conditions of the described experiment, high frequency
energy absorption by a plasma close to o) = w^ is "collision absorption"
in the sense that it is described by (3), where f(X) has the form of (4).
Formulas (3) and (4) may be used to determine the fact that, in the case
of a hydrogen plasma, the absorbed power is at a maximum in the case of
^=^ = 2.44.10-'7XW,-. (5)
Ho
where 5 = — ; Hq is the magnetic field corresponding to cyclotron
15
15 NIO%-'
resonance of ions at the oscillator frequency; B^ — the magnetic field
forwhich the high frequency power absorbed by the plasma is at a maximum.
The dependence described by (5) is shown in Figure 1 (straight line) .
This figure also presents the points computed on the basis of experimental
data regarding the shift in the absorption maximum from Hg for different
plasma densities measured by an interferometer for a line current of
4.5 a/cm. Over the entire range of measured plasma densities, the devia-
tion of the points obtained experimentally from the computed dependence
does not exceed the limits of measurement errors.
In order to determine what collisions cause the observed high fre-
quency energy absorption, a study was made of the dependence of the
effective frequency of ion collisions on the neutral gas pressure and on
the plasma electron density. It can be seen from (3) and (4) that the
effective frequency of ion collisions Ypff can be expressed either as
1 e
— • — — AH — where AH is the halfwidth of the resonance absorption curve
2 Mc
AXUi
— or as
.2
jg, where A is the factor in front of jgfCX) in (3); S^ —
m
the power at the absorption maximum.
It was shown in (Ref . 1) that the power at the absorption maximum
and the width of the resonance absorption curve depend slightly on the /21
neutral gas pressure in the 0.133 - 1.33 n/m^ range, due to which fact
the observed power absorption cannot be caused by ion collisions with
neutrals .
Figure 2 presents the dependence obtained experimentally of the
effective frequency of ion collisions Ypff on the plasma electron density
for a line current 4.5 a/cm. The crosses designate the points computed
16
Y<
= f
fio'^
3« C
Fi
1
^
-
3
^
-
2
^V,
~v
e
Tl-
•
/
as
Ofi
a^
0.2
. —
—
—
-
1
/
1
^
T
60U
.microsec
1
\
500
400
^ 300
1
V
--
-
ZOO
100
■r^
*-<
0//'34S6739 /a,a/c«
Figure 3
according to the halfwidth of the absorption curves; the dots designate
the points computed according to the absolute power at the absorption maxi-
mum for a given density. The observed high frequency energy absorption is
caused by ion-electron collisions. However, under the conditions for
which the dependence shown in Figure 2 was determined, the electron tem-
perature comprised 1 ev in order of magnitude (see Figure 3) . The frequency
of ion Coulomb collisions with electrons must be one order of magnitude
less than Ypff over the entire range of measured plasma densities. There-
fore, the effective frequency of ion collisions, which was measured in the
experiment, cannot be caused by Coulomb scattering of ions by electrons
[the electron temperature, which was determined in (Ref . 1) according to
Y ff under the assumption of Coulomb interaction of ions with electrons, /22
amounted to 0.15 ev) .
In order to clarify the nature of the interaction between ions and
electrons, and consequently the nature of the observed high frequency
energy absorption by a plasma in ion cyclotron resonance, the dependence
of the effective frequency of ion collisions on the line current density
JO — i.e., the dependence on the strength of the high frequency field in
the plasma — was determined. The quantity Ygff ""^^ computed according to
17
the power at the absorption maximiun for a density of 5.1 •10-'^^ cm ^.
Figure 3 shows the dependences Ygff (Jo)» '^^(■20^ a^^i t(jo) — where t is
the plasma decay time — on density 7. 6*10^^ up to 2.5'10-'-^ cm~^ for a
magnetic field strength of 4.2»10^ a/m, where the absorption reaches a
maximum for a density of 5.1*10^^ cm"^. It can be seen that Yoff rapidly
decreases with an increase in joCjo*^ 1 a/cm) . The frequency of Coulomb
collisions between ions and electrons must follow this pattern, since the
high frequency power introduced into the plasma — and consequently the
electron temperature — increases with an increase in j q . The electron
temperature, determined according to Yeff ^^ ^^^ case of Jq< 1 a/cm under
the assumption of Coulomb interaction between ions and electrons, coin-
cides in order of magnitude with the temperature computed according to
the line intensity H (see Figure 3) .
p
There is a sharp bend in the curve y ff (Jo) ^^ the point Jo ^ 1 a/cm,
and with a further increase in jg, y^ff increases slowly, remaining at a
level on the order of 2*10^ sec~^ and considerably exceeding the frequency
of Coulomb collisions between ions and electrons for given electron tempera-
tures. As can be seen from Figure 3, at this point a sharp decrease in the
plasma decay time occurs. It was shown in (Ref . 2) that a decrease in the
decay time is caused by increased plasma diffusion across the force lines
of the magnetic field. In its turn, the diffusion is caused by an insta-
bility produced in the field of an ion cyclotron wave having a large ampli-
tude.
It may thus be assumed that under the conditions of our experiment
the anomalously large frequency of ion collisions in high frequency fields
with supercritical strengths is caused by the more intense (as compared
with Coulomb interaction) interaction of ions with electrons. The reason
for this (and for anomalous diffusion) is "bunched" instability. The fact
that absorption is "collision absorption" — i.e., it fomnally satisfies
equation (3) — where f (x) in the form of (4) means in physical terms that
the damping force (which is caused by an instability) of the ion motion /23
directed toward the electrons is proportional to the relative velocity of
ion and electron liquids in the field of an ion cyclotron wave.
Comparison With the Theory of ^on
Cyclotron Wave Stabi l i ty^
We shall show that the values of the effective frequency of ion colli-
sions in fields with supercritical strengths, computed either according to
the halfwidth of the resonance absorption curve or according to the absolute
power at the absorption maximum, as well as the plasma diffusion coefficient
determined by the decay time, coincide in order of magnitude with the corres-
ponding values computed for our case according to the theory of ion cyclotron
wave stability (Ref. 8).
18
As follows from (Ref. 8), under the experimental conditions (Ng~
~5*10^2 cm~^; H~ 4*10^ a/m; Tg ~ 1 ev) for longitudinal (electrostatic)
high frequency fluctuations excited by an ion bundle and propagated almost
across a constant magnetic field (cos 9 :$, 2.) , we obtain
Reu) — Imo) — (u>e"Jj)'^'> (6)
where o) is the cyclotron electron frequency.
The effective frequency of ion collisions may be regarded as the in-
verse of the time during which an ion bundle with an initial energy per
unit of volume N^ — r— (u — the velocity of the relative motion of the ion
and electron components in the field of an ion cyclotron wave almost equals
the velocity of an ion liquid) excites the fluctuations (6) and transmits
all of the energy of the ordered motion to the electron gas. On the other
2
hand, the electrons obtain the energy Ni from the ions during the ex-
bundle is thus
citation time of the fluctuations (u oj.) '^. The braking time of an ion
-:=— (u),o),) 2. (7)
m
Substituting numerical values in (7), we obtain 10 ° sec for x, which coin-
cides in order of magnitude with Ylf f ~ 0.5'10~^ sec which was determined
experimentally.
Let us determine the diffusion caused by instability, employing the /24
theory of nonuniform plasma stability (Ref. 14). The diffusion coefficient
may be written as follows
D^V^(, (8)
where V is the plasma pulsation velocity; t = — — the characteristic time
of correlation disappearance. The increasing increment c£ fluctuations (6)
must be used as v. The pulsation amplitude may be determined by the condi-
tion of balance between two processes — one of which leads to an increase
in the pulsation amplitude due to the development of instability, and the
other leads to contraction of this amplitude due to nonlinear processes
leading to oscillation damping. As a result, we obtain
y = vx^, (9)
where Xj_ is the oscillation wavelength in the radial direction. Then the
19
diffusion coefficient is
£>~vXi, (10)
or, since ku ss v, where k is the radial wave number, we have
D~(2u)«^. (11)
Under the experimental conditions, in the case of jo = 8 a/cm, u ~
~ 10 cm/sec and D ~ 2'10 cm^/sec, which coincides in order of magnitude
with D ~ 5 '10^ cm^/sec, determined for the same conditions according to
the plasma decay time in (Ref. 2).
Conclusions
1. The absorption of high frequency energy, which was observed in
our experiments, close to the ion cyclotron resonance both for small and
large amplitudes of the high frequency field can be formally described by
relationships (3) and (4), which were derived for the case of "collision"
absorption.
2. Absorption is caused by the interaction between plasma ions and
electrons. In fields whose strength is less than the critical strength,
it is caused by pair Coulomb collisions. In fields whose strength is
greater than the critical strength, the effective frequency measured
experimentally of ion collisions is considerably greater than the fre-
quency of pair Coulomb collisions. In this case, the anomalously large /25
absorption is apparently caused by braking of the ordered motion of ions
in the field of the ion cyclotron wave by high frequency longitudinal
oscillations excited by an ion bundle.
3. The value obtained experimentally for the effective frequency
of ion collisions, and also the diffusion coefficient determined according
to the plasma decay time, coincide in order of magnitude with the corres-
ponding values calculated according to the theory of ion cyclotron wave
stability.
4. On the basis of the results obtained, it is natural to pose the
question of "turbulent" plasma heating (Ref. 15) in ion cyclotron resonance
— i.e., brief excitation in the plasma of an ion cyclotron wave having a
large amplitude, with subsequent thermalization of the ordered motion of
plasma particles in the field of this wave by the high frequency longi-
tudinal oscillations excited by an ion bundle passing through the electron
gas. The heating period must be quite small, so that anomalous diffusion
caused by plasma instability does not produce significant losses in plasma
particles.
20
REFERENCES
1. Vasil'yev, M. P., Grigor'y^va, L. I,, Dolgopolov, V. V., Smerdov, B. I.,
Stepanov, K. N. , Chechkln, V. V. Zhurnal Tekhnicheskoy Flzlki, 34,
6, 1964.
2. Chechkin, V. V., Vasil'yev, M. P., Grlgor'yeva, L. I., Smerdov, B. I.
Yadernyy Sintez, 4, 2, 1964.
3. Sagdeyev, R. Z. , Shafranov, V. D. In the Book: Plasma Physics and
the Problem of Controlled Thermonuclear Reactions (Fizika plazmy 1
problema upravlyayemykh termoyadernykh reaktsiy) . Izdatel'stvo AN
SSSR, Moscow, 430, 1958.
4. Stix, T. H. Phys. Fluids, 1, 308, 1958.
5. Vasil'yev, M. P., Grlgor'yeva, L. I., Dolgopolov, V. V., Smerdov,
B. I., Stepanov, K. N., Chechkin, V. V. Zhurnal Tekhnicheskoy
Flziki, 6, 34, 1964; In the Book: Plasma Physics and Problems of
Controlled Thermonuclear Synthesis, 3 (Fizika plazmy i problemy
upravlyayemogo termoyadernogo slnteza, 3). Izdatel'stvo AN USSR,
Kiev, 96, 1963.
6. Kadomtsev, B. B. In the Book: Plasma Physics and the Problem of
Controlled Thermonuclear Reactions (Fizika plazmy i problemy uprav-
lyayemykh termoyadernykh reaktsiy), 4. Izdatel'stvo AN SSSR, Moscow,
364, 1958.
7. Kurilko, V. I., Miroshnichenko, V. I. In the Book: Plasma Physics and
Problems of Controlled Thermonuclear Synthesis, 3 (Fizika plazmy i
problemy upravlyayemogo termoyadernogo sinteza, 3). Izdatel'stvo
AN USSR, Kiev, 161, 1963.
8. Stepanov, K. N. Zhurnal Tekhnicheskoy Fiziki, 34, 12, 1964.
9. Hooke, W. M. Nuclear Fusion, Supp., 3, 1083, 1962.
10. Nazarov, N. I., Yermakov, A. I., Tolok, V. T. Present Collection, 1.
11. Nazarov, N. I., et al. Plasma Physics and Problems of Controlled /26
Thermonuclear Synthesis, 3 (Fizika plazmy i problemy upravlyayemogo
termoyadernogo sinteza, 3). Izdatel'stvo AN USSR, Kiev, 164, 1963.
12. Dubovoy, L. V., Shvets, 0. M. , Ponomarenko, A. I. In the Book: Prob-
lems of Magnetic Hydrodynamics (Voprosy magnitnoy gidrodinamiki) 2.
Izdatel'stvo An Latviyskaya SSR, Riga, 355, 1962.
13. Yakovlev, K. A., Pankrushina, D. K. , Basin, Yu. G. Pribory 1 Tekhnika
Eksperimenta, 4, 89, 1961.
21
14. Galeyev, A. A., Moiseyev, S. S., Sagdeyev, R. Z. Atomnaya Energiya,
15, 6, 451, 1963.
15. Zavoyskiy, Ye. K. Atomnaya Energiya, 14, 1, 57, 1963.
INVESTIGATION OF CONDITIONS PRODUCING A DENSE PLASMA IN A
METALLIC CHAMBER AND ITS HIGH FREQUENCY HEATING
0. M. Shvets, S. S. Ovchinnikov, V. F. Tarasenko,
L. V. Brzhechko, 0. S. Pavlichenko, V. T. Tolok
It was shown in (Ref . 1) that powerful, high frequency oscillators may
be employed to produce a dense plasma in a metallic chamber. The present
experiments represent a continuation of this study by investigating the
conditions producing a dense plasma in a metallic chamber and by determining
the main plasma parameters.
The utilization of a metallic chamber as the operational vacuum body
and the method of feeding the system from the hf oscillator have definite
advantages and unusual features.
1. The system has a low input impedance, which does not require high
voltages when large, high frequency powers are introduced, and avoids
several technical difficulties which accompany, as an example, the spatially
periodic circuit proposed by Stix (Ref. 2).
2. A good connection between the feed electrodes and the plasma is
provided. A dense plasma with a cylindrical form is produced between the
central electrodes, and has no direct contact with the wall of the discharge
chamber.
Investigation of the Conditions Producing a
Dense Plasma in a Metallic Chamber
The "Vikhr"' device (Figure 1), on which the study was performed, con-
sists of a copper chamber 1 with a wall thickness of 2.5 mm, an inner diameter
of 125 mm, and a length of 2000 mm. It is placed in a magnetic field which 111
can be regulated continuously between -2*10^ a/m. The configuration of the
magnetic field may be varied, depending upon the method chosen to handle
the plasma produced. Since we were interested in plasma heating by high
22
Figure 1
frequency fields , the magnetic field configuration was chosen in accordance
with the conditions which were requisite for generation and absorption of
ion cyclotron waves. The region of the "magnetic beach" was located in
the center of the selenoid 11 producing the magnetic field. The residual
gas pressure in the system did not exceed 1.33-10"'* n/m^. The rods 3 are
introduced axially into the vacuum chamber through the porcelain insula-
tors 2. Aluminum electrodes 4 with a diameter of 50 mm and a length of
70 mm are located at the ends of these rods. The distance between the
electrodes is 1000 mm. The feed rods 3 through the coaxial cables 5 and
the capacitance C are connected with the coupling coil 6 of the hf oscil-
lator 7. The oscillator power is on the order of 100 kw; the operating
frequency is 1.82' 10^ cps. The ends of the chamber are closed with glass
discs 8. The operational gas from the flask 9 is admitted into the chamber
by the valve 10 through an opening in the glass disc 8.
With the system employed for switching on the hf oscillator, the feed
electrodes acquire a negative potential, due to the rectifying properties
of the plasma; this negative potential can reach several kilovolts with
respect to the chamber. This potential creates the condition for oscilla-
tion of electrons along the magnetic field force lines between electrodes,
similarly to Penning discharge. The oscillating electrons effectively /28
ionize the operational gas and provide for a high degree of ionization up
to a chamber pressure of 1.33*10-^ n/m^. A core of dense plasma is formed
between the electrodes along the chamber axis; the diameter of this plasma
is determined by the electrode diameter. Due to the presence of a constant
potential, the peripheral plasma begins to rotate according to the law of
plasma behavior in crossed electric fields and magnetic fields. In a few
microseconds the applied high frequency voltage produces a discharge in /30
the operational area, which is accompanied by a voltage decrease at the
electrodes due to an increase in the oscillator loading (Figure 2 , a) .
The central electrode acquires a negative potential, which is retained for
the entire period of time. that the dense plasma exists (Figure 2, b) .
23,
w ■
9n i
msec
Figure 2
Lines of admixtures from the chamber wall materials and from the
electrodes — Cull, AIII — and a weak line CII (Figure 3) were observed
in the discharge spectrum obtained by means of a ISP-51 spectrograph with
a UF-85 camera (focal distance 1300 mm) .
When the outline of the line H„ was measured, it was found that under
p
our experimental conditions there was an apparant Stark widening by the
micropoles of the plasma. This made it possible to determine the charged
particle density, by comparing the contour observed experimentally with
the theoretical contour computed on the basis of the theory advanced by
Kolb, Grim, and Shen (Ref. 3). For purposes of comparison. Figure 4 shows
the observed contour I and the theoretical Stark contour II , computed for
n = 2.0*10^'* cm~^. The figure also plots the Gaussian contour III with
the halfwldth equalling the experimental value of the halfwidth 0.7 A.
Close to the maximum, the line broadening was caused by the Doppler
mechanism, and the slopes of the line are due to broadening by the Stark
24
/29
He I 5875
Cul 5216
Cui 5153
Cul 5111
Cui 5101
Hei 50^7
Hei 5015
Hei mi
H^ mi
Cll mtf
Hei ^713
Hen me
Hi ttd^O
Figure 3
mechanism.
Measurements of the plasma density made it possible to determine its
comparatively weak dependence in discharge on the magnetic field strength,
which confirmed the assumption regarding the nonresonance mechanism by
which a plasma is produced in discharge. At a pressure of 0.4 n/m^ , the max-
cm^
imum plasma density was 2*10^
For an approximate determination of
25
Figure 4
the plasma density distribution over the radius of the system, different
sections of the plasma were focused on the spectrograph slit. The most
>1^ 1
dense plasma (density on the order of 10^
cm
3" ) produces a filament with a
diameter of 20 mm; at a distance of about 30 mm from the discharge axis,
the plasma density decreased by more than one order of magnitude.
The electron temperature was measured with respect to the intensities /32
of singlet and triplet lines of helium; it was (4 - 5) 10 °K.
The dependence of the plasma density change on time was determined
by probing the plasma with an ultrahigh frequency signal at a wavelength
of 3 cm (Figure 5, c) and 0.8 cm (Figure 5, d) through the glass openings in
the chamber. The measurements showed that in an optimum regime there is
a plasma with a density of > 10 — j in the apparatus for 3.6 milliseconds.
cm-
and with a density of > 10
12
cm^
for 17 milliseconds. The form of the hf
pulse is shown in Figure 5, a. The maximum plasma density was determined
according to measurements of the relative change in the intensity of the
spectral line H. with time. The intensity of the spectral line H„ (the
p p
photomultiplier signal. Figure 5, b) , if the main excitation mechanism is
electron collision, is
h
•^4 2 + ■^41
where v^2 is the frequency corresponding to the line H ; A., — probability
of spontaneous transition from the i level to the k level;
< vai (u) } = f voi {v) fe {v) dv;
26
/31
w
. — -> ,-,.
i^U t, msec
Figure 5
a. (v) — excitation cross section (by electron collision) of the n = 4
level. fgCv) is the electron distribution function with respect to
velocity; [v^^Cv)] in the case of Maxwell distribution of electron veloci-
ties is the function of electron temperature which — as measurements of
the dependence T (t) have shown — changes very little during the dura-
tion of the hf pulse. Thus, the intensity of the line Hg must change
proportionally to the product n n In our case, the neutral gas from
the cold section constantly enters the discharge column, since
Rpl
^n
10~^ sec (Rpl — radius of the plasma coliimn; v — thermal velocity
27
of the neutrals). Therefore, the neutral density n^ « const and, conse-
quently, the intensity of the line H is proportional to the electron
p
density n . At the moment when n = 10-^^ — g-, the intensity of the line
H decreases by a factor of 200 as compared with its maximum intensity —
i.e., the maximum plasma density in our case is 2*10-^'*^ — j, which coin-
cides with the contour of the line H^ determined according to Stark
p
broadening. The plasma density determination based on the intensity of
the line H , at the moment when a signal with a wavelength of 0.8 cm /33
begins to pass through, coincides with the result of microwave measure-
ments .
Thus , we have studied the conditions producing a dense plasma in a
metallic chamber. A plasma with a density on the order of 10^** — r
■' cm^
and an electron temperature (4 - 5) 10^ °k was obtained in the experiments.
The weak dependence of plasma density on the magnetic field strength points
to a nonresonance mechanism by which the plasma is produced. The aim of
our subsequent experiments was to investigate the heating of the plasma
obtained by the generation of ion cyclotron waves.
High Frequency Heating of a Dense Plasma
in a Metallic Chamber
The next stage of our investigation was to study the possible heating
of ions by generating ion cyclotron waves in a dense plasma produced by a
hf oscillator in a metallic discharge chamber. The "Vikhr"' device was
modernized so that, when the ion cyclotron wave was generated at the ends
of the coaxial close to the magnetic mirror, it was propagated into the
center of the discharge chamber, where the magnetic field strength was de-
creased to a value equalling the cyclotron value for protons, forming the
region of the magnetic beach. When the wave was propagated along the system
axis and approached the region of the magnetic beach, its velocity decreased,
and it was damped, transmitting its energy to the plasma ions. In order
to propagate the wave in the region of the magnetic beach, it is necessary
that the magnetic field strength per wavelength change by several percents
— i.e., a smooth change in the magnetic field strength over the length of
the system is requisite. If the opposite is true, the wave will be reflected.
The resid-ual gas pressure in the system did not exceed I.S'IO"** n/m^.
The experiment was performed with a mixture of two gases — hydrogen and
helium or hydrogen and argon. The operational pressure was established
by a stationary regime of the valve operation in the 0.1 - 0.8 n/m^
range. At a pressure of 0.4 n/m^, the hf oscillator, operating at a
28
frequency of 1.82 '10^ cps with a power of 150 kw, produces a plasma with
a density on the order of 10-^^ j. With a magnetic field strength of
H = 1.25 H. (H. — the cyclotron value of the magnetic field strength
x.c i.c
A.
for protons) waves are generated in the plasma with X = = 6.6 cm /34
(A — wave length in a vacuum: n= f ' i o> \2 — refractive index
vac ^ ' , / - ( 1
i; n= f ' / 0. \2
TJ
of the medium; v. = -—^^^ — Alfven velocity; H — magnetic field; u) —
operational frequency of the oscillator; w. — cyclotron frequency of
proton rotation; p — mass density.
This system has several advantages. The pulse which is initially
transmitted to the ions is perpendicular to the vector of the outer mag-
netic field strength, which facilitates the retention of ions in the cork-
screw configuration of the magnetic field. Since the system has a low inped-
ance, energy is readily introduced into the discharge chamber and is trans-
mitted directly to the plasma ions. An increase in the density and diameter
of the plasma does not make the conditions worse for wave generation. At
the same time, the spatially periodic circuit for Introducing hf power
into the plasma, which was advanced by Stix (Ref. 2), loses any physical
meaning with an increase in the plasma density and diameter. When a mix-
ture of two gases is heated, or when it is necessary to heat the plasma
electron component simultaneously with the ion component, this system makes
it possible to introduce the power of two oscillators operating at different
frequencies.
As has been pointed out , an Increase in the plasma density does not
impede wave generation, since spatial periodicity is not given externally,
but is established as a function of the plasma refractive index, and may
be small (several centimeters). However, the possibility of periodicity
is not excluded, if rings are placed endwise on the coaxial at different
distances from each other; these rings will introduce a perturbation,
creating a specific periodicity along the system axis. One unusual feature
of the discharge is the fact that its nucleus, consisting of a plasma which
is almost entirely ionized, is surrounded by a plasma having a low density
and a neutral "housing" . During generation and absorption of ion cyclotron
waves, the dense plasma nucleus is heated, and the cold plasma surrounding
it with a low density contributes, in all probability, to the suppression
of channel instability.
In the experiments described, the ion temperature of the plasma and
the charged particle density, which were averaged over time, were deter-
mined by means of optical methods. The change with time in the intensity
of the spectral lines for hydrogen and admixtures was also studied. A
ISP-51 spectrograph was placed in such a way that the central portion of /35
29
^^^^''d-
Figure 6
the cylindrical plasma column, lying in the region of the magnetic beach,
was focused on its input slit. Oscillograms showing the intensity of
the spectral lines for hydrogen and helium or argon, added to the chamber
in small proportions to the operational gas, showed that regular oscilla-
tions are produced in the intensity of the spectral lines with a frequency
on the order of 20 kc, close to the magnetic field strength corresponding
to the cyclotron value for protons (Figure 6) . These oscillations appear
during wave generation in the plasma, and are caused by the eccentric ro-
tation of the dense plasma filament as a whole with respect to the chamber
axis, in accordance with the drift law in crossed, radial electric fields
and axial magnetic fields. The direction and frequency of the filament rota-
tion was determined by two photoelectron multipliers oriented towards the
end of the chamber and placed at a radius of 3 cm from the system axis.
One of them was shifted along the azimuth. The oscillation phase of the
light intensity was thus changed.
One interesting feature was discovered when the ion temperature was
measured according to the Doppler broadening of the hydrogen and additional
gas lines . The width of the hydrogen line depended comparatively little
on the magnetic field strength, and the additional gas lines were broadened
considerably when the magnetic field strength was close to the cyclotron
value for protons. The ion temperature of this gas, determined for the /36
optimxun operational regime of the apparatus, amounted to 250 ev (2.5*10^ °K) .
Measurements of the halfwidth of the line H. showed that the hydrogen atom
temperature was below the temperature of the additional gas, while there
was considerable Stark broadening of the line contour, corresponding to a
ai+ L
plasma density of 2*10
cm-
There was also Doppler broadening of the
admixture lines (copper, alvuninimi, oxygen, carbon, nitrogen). When the
temperatures of different admixtures and additional gases were measured.
30
l-'t
■* 8 12 16 H-10":alt^i
Figure 7
no Ion temperature dependence on their mass was detected. The heating
of the additional gas ions was terminated when only one additional gas
was admitted into the discharge chamber up to the previous operational
pressure. One of the characteristic dependences of helium temperature
(when it was added to the chamber in a small amount) on magnetic field
strength, measured over the half width of a helium line with a wavelength
of 4921.93 A, is shown in Figure 7. The helium atoms acquire a maximum
temperature at a magnetic field strength which is close to the cyclotron
value for protons. A certain temperature increase is observed as a mag-
netic field strength is approached which equals the double cyclotron value
for hydrogen ions.
If it is assumed that the halfwidth of the H_ line is determined by
p
Doppler broadening, the temperature of neutral hydrogen is considerably
lower than the plasma ion temperature. However, under our conditions
(in the case of T = [4-5] 10^ °K) , the lifetime of neutral hydrogen in
the plasma, with respect to the ionization process, was small as compared
with the time of Coulomb collisions. Therefore, neutral hydrogen cannot
acquire energy equalling the ion energy. Since the frequency of hydrogen
atom collisions with electrons is greater under these conditions than the
frequency of collisions with ions , the Stark mechanism is the predominant
mechanism leading to the broadening of the line of the residual neutral
hydrogen.
In an optimimi operational regime in a mixture of two gases — hydrogen
and argon, the dependence of the argon ion temperature on the magnetic
field strength is resonant in nature with a maximirai close to the magnetic
field strength corresponding to the cyclotron value for protons . The
maximum temperature of argon ions in the experiments was 2. 5 •10° °K, and
for electrons — 5 '10^ °K. In order to determine the distribution of the
plasma ion temperature over the radius of the system, different sections
of the plasma were focused on the spectrograph slit. The hottest plasma was
located in the center of the filament. The temperature rapidly decreased /37
over the radius (approximately 5 times greater along the axis than on a
radius of about 4 cm) . Figure 8 Illustrates the temperature dependence of
31
' '.5 2 U.kv
Figure 8
plasma on the applied high frequency voltage. The increase in the mag-
netic field strength and the high frequency voltage — i.e., the hf power
introduced — made it possible to obtain a higher plasma temperature.
The mechanism by which the proton energy is transmitted to the addi-
tional gas has still not been definitely clarified. Since the gasokinetic
pressure in these experiments may exceed the magnetic pressure, the possi-
bility is not excluded that centrifugal instability may be produced, which
can lead to transmission of proton energy to the additional gas and can
lead to its heating.
In certain operational regimes of the apparatus, we observed genera-
tion of a rapid magnetosound wave at a magnetic field strength which was
less than the cyclotron value for protons. However, conditions were not
favorable for studying it at the existing oscillator frequency (1.82-10°
cps) .
Thus, in all probability, these experiments illustrate the feasibility
of high frequency heating of a dense plasma consisting of two types of
ions by resonance generation of ion cyclotron waves for one type of ions.
The mechanism by which energy is transmitted from one type of ions (pro-
tons) to other ions (helium, argon, admixture) requires further study.
REFERENCES /38
1. Shvets, 0. M. , Tarasenko, V. F., Ovchinnikov, S. S. , Tolok, V. T.
In the Book: Plasma Physics and Problems of Controlled Thermonuclear
Synthesis (Fizika plazmy i problemy upravlyayemogo termoyademogo
sinteza), 3. Izdatel'stvo AN USSR, Kiev, 117, 1963.
2. Stix , T. and Palladino, R. In the Book: Transactions of the Second
International Conference on the Peaceful Utilization of Atomic
Energy (Trudy Vtoroy mezhdunarodnoy konferentsii po mirnomu ispol'-
zovaniyu atomnoy energii) . Geneva, 1958. Selected Reports of
32
Foreign Scientists. Physics of a Hot Plasma and Thermonuclear
Reactions (Izbrannyye doklady inostrannykh uchenykh. Fizika gorya-
chey plazmy i termoyademyye reaktsil.) Atomizdat, Moscow, 242,
1959.
3. Grim, H. R. , Kolb, A. C, Shen, K. J. Stark Broadening of Hydrogen
Lines in Plasma, Phys. Rev., 116, 4, 1959.
HIGH FREQUENCY PLASMA HEATING
K. N. Stepanov
As is well known, a rapid decrease in the frequency of Coulomb colli-
sions of plasma particles leads to a decrease in the rate of ohmic plasma
heating both by constant fields and by high frequency electric fields when
there is a temperature increase. Therefore, great hopes have been expressed
(Ref. 1, 2) for achieving plasma thennonuclear temperatures by utilizing
different effects of collislonless energy absorption of high frequency
fields by a plasma. In principle, this amounts to Cherenkov, or to cyclo-
tron, absorption and radiation of waves by electrons and ions.
However, collislonless energy absorption by a plasma leads to great
distortion of the velocity distribution function of electrons and ions and
to attenuation, or even complete discontinuance, of absorption. It also
leads to every type of plasma instability. If collislonless plasma heating
is effectuated by weak electric fields, so that Coulomb collisions provide
a Maxwell distribution function, the heating rate is the same as for ohmic
heating. The time required to heat the plasma up to thermonuclear tempera-
tures with weak fields is less than the time required for containing the
plasma in a thermonuclear reactor, utilizing the reaction D + D, with a
positive energy output. Therefore, under the condition of producing a
stable, plasma configuration, it is primarily possible to achieve thermo-
nuclear plasma temperatures with slow heating by weak, constant fields or
by high frequency fields, which in themselves do not lead to strong plasma /39
instability.
On the other hand, for rapid plasma heating it is very tempting to
employ strong electric fields, under whose influence electrons or ions
in the plasma acquire a large directed velocity (Ref. 3-6), as well as
strong electron bundles (Ref. 7) or the collision of plasma clusters
(Ref. 3) ("turbulent" methods of plasma heating). Due to the development
33
of bunched Instability in the systems, there is a rapid increase in the
energy of high frequency oscillations leading to braking of the acceler-
ated particles, to an energy increase with respect to motion ("tempera-
ture"), and to an energy exchange between different plasma components.
This article presents a brief survey of the collisionless, high fre-
quency methods of plasma heating, and it also compares them with the
ohmic heating method.
Ohmic Plasma Heating
When investigating the processes of plasma heating, we shall disre-
gard energy losses due to cyclotron radiation of electrons, which is ab-
sorbed by a plasma if its dimensions are fairly large, as well as losses
due to plasma thermal conductivity on the chamber walls — which is also
a surface effect. In addition, we shall assume that the residual gas
pressure is small, so that we can disregard energy losses due to over-
loading and radiation of admixed atoms excited by electron collision.
Under these conditions, the energy losses by the plasma are determined
only by braking radiation of the electrons .
When the ohmic method is employed to heat a plasma by a "constant"
electric field with the strength E, the electron energy increase is deter-
mined by the following equation
§=^Q+-Q-, (1)
3
where w = y T is the mean electron energy; T — electron temperatures;
n,Q+=aE^ (2)
— Joule heat liberated per unit volume per unit of time; ng — electron
density; a = _^ e — plasma conductivity; ^ = ^ '"'''' — time of /40
electron mean free path; A — Coulomb logarithm. The intensity of elec-
tron braking radiation (Ref . 8) is
32 yW.e«
Q— = - - a)
If we take into account the transmission of energy to ions , then we
must replace Q, by -r Q, . in the case of T^ « T^. This effect, as well as
other effects which do not change in order of magnitude, is not taken
into account from this point on.
34
Under the Influence of the electric field E, the electrons acquire
a directed velocity
u =
enoVg
If the field strength E exceeds the critical value (E ~ , where
a
y«=l/ ^ is the electron thermal velocity), then all of the electrons
are carried along by the electric field in a continuous acceleration regime.
The motion of the electron gas with respect to the ions leads to the phe-
nomenon of bunched instability which is related to the buildup of longi-
tundinal high frequency oscillations in the plasma (Ref. 9). The inverse
influence of plasma oscillations on the electron motion leads to electron
braking — i.e., to anomalous plasma resistance, and also to increased
radiation of radio waves, which considerably increases the plasma thermal
radiation (Ref. 10, 11).
Plasma bunched instability can arise in the case of E << E^^, i.e.,
in the case of u << Vg. For example, in a very non-isothermic plasma,
in the case of T^ >> Tj^ (T- — ion temperature) the electrons build up
sound oscillations (Ref. 12), if the velocity u exceeds the speed of sound
Vs = l/ ^ . In the case of T < T^, the electrons build up longitudinal
ion cyclotron oscillations, if u > 10 (v^ = l/ — ■ — thermal ion velo-
city) (Ref. 13). However, it may be expected that — since only a small
group of resonance electrons participates in the oscillation buildup —
the formation of a "plateau" In the electron distribution function will /41
lead to a decrease in the Increasing increment ( Ref. 14 - 18), and nonlinear
effects will lead to stabilization of these oscillations (whose amplitude
will be small) (Ref. 16 - 18).
Expression (2) may be employed only in the case of E << E^^.. Assuming
that E = aE^j-, where a << 1, we obtain
Q+ ^' (4)
Let us substitute expressions (3) and (4) in (1) . We then have
dt
V T.
5-lG-'5Kr,jnokev/sec. (5)
35
where Tg — in kev, ng — in cm ^. The heating process terminates,
-J— = in the case of
T —S-lO^a^kev
max . ,,^
It thus follows that even in the case of heating with weak fields (a =
= 10" -"^ - 10~^) it is possible to achieve thermonuclear temperatures
(T -x. 50 kev) .
If Q_ << Q, , then — disregarding Q as compared with e Q in (1)
and taking into account (4) — we obtain
n = 7-.(i + "^)"'-. (7)
where Tq and tq — are the initial values of T^ and x . The influence of
braking radiation on the heating process is significant in the case of
Tg 0, Vax' i*^*» ^^ the case of
/5- WV/i
^-^"^[-f^j ■ (8)
For example, in the case of Tg '\. 100 ev, ng 'v 10^^ cm~^, Tg 'v^ 2"10 ^
sec and a '^ (1/30), the temperature T = 50 kev is achieved during the
time t '^ 0.2 sec
'-AtJ ' (9)
and the electric field strength changes between 0.3 - 3 '10"^ v/cm. The
energy exchange between ions and electrons takes place during the time
T^g ~ 0.5 sec
mi
^'•«~^^m, • (10)
so that the separation between the electron temperature and the ion tem- /42
perature is small.
The heating time (9) is not large from the point of view of producing
thermonuclear reactors with a positive energy balance. The time the plasma
is contained in such a reactor, when employing the reaction D + D, must be
greater than (Ref. 19)
t*—-^ sec. (11)
36
For the example under consideration (ng '^ 10-^^ cm~^) t* "^ 10 sec,
which exceeds the heating time of the Ion component by a factor of 20.
Cherenkov Ion Heating
In the case of high frequency plasma heating employing the method
advanced in (Ref. 20), oscillations of an axial magnetic field H^ which
is produced by azimuthal electric currents, lead to a variable azimuthal
electric field E, in a plasma cylinder located in a strong longitudinal
field Hq. Radial, drift oscillations of the plasma arise, due to particle
drift in crossed fields E. and Hq, which leads to density oscillations —
i.e., to the appearance of sound (more precisely, magnetosound) waves.
(It is assumed that the wave frequency w is considerably less than the ion
gyrofrequency ui^ = 2.^ and the wave length is considerably less than the
m-j^c
V-f
Larmor radius of ions p,- = with thermal velocity).
wi
Ions having the velocity v,, along Hg , which is close to the phase
velocity of a wave Va H , vigorously interact with the field E. . If
•^ II
the wave phase velocity is on the order of the ion thermal velocity, the
number of resonance ions is large, and there is strong wave absorption.
(As is known, in the absence of a magnetic field sound oscillations in a
plasma, in the case of Tg < Tj_, cannot be propagated in general, due to
strong Cherenkov absorption by ions (Ref. 21); in the case of Hq ^^ and
Tg < Tj^, magnetosound oscillations are also damped during one period, if
V^ ^Vi [Ref. 22]).
Let us investigate Cherenkov absorption by plasma ions of the energy
of an electromagnetic field produced by azimuthal electric currents, /43
which take the form of a moving wave and flow into the coll placed on a
plasma cylinder having the radius a:
/^=/„cos(;%,z — W)8(/- — a). (12)
If k|| a< 1, then the current (12) produces a variable longitudinal
magnetic field with the strength
H^=Hcos{k,z — wt), (12')
where H = — il. The strength of the aximuthal electric field is
ff
Ef = — ^— kirs'm{k,z — uif).
knc
2n,
where ni = is the longitudinal refractive index.
37
In the absence of damping, the value of the energy flux, averaged
over time, in the plasma per unit length
•^0 "^ "■ 4^ Ey/ZzSica
equals zero. It is apparent that, in the presence of damping, the energy
flux in the plasma equals, in order of magnitude,
4«," in^y (13)
where x is the correction to the "transverse" refractive index nj_, caused
by the Cherenkov oscillation absorption by plasma ions.
The dispersion equation for a magnetosound wave has the form (Ref . 22)
^/, , .-2 "f
"a
<- + ^\ =^M+'«-P^' (14)
where the coefficient n ~ 1, if Vx ~ v^. The component ~ i takes into
account Cherenkov wave damping in an ion gas .
Let us investigate a plasma with a small gasokinetic pressure
/SitnoT,-,, \ ff
— 772~^C1 • In this case, the Alfven velocity Va=-t== is consider-
\ "o J y4jinom..
ably greater than the ion thermal velocity v^^ and the speed of sound Vg.
Since nn ~ — , in the zero approximation it follows from equation (14) that /44
nj_ = in||. In the following approximation we obtain: "j. =ini (l + y '"^j ^^)'
I.e.
Taking into account expressions (13) and (15) , we obtain the following
formula for the mean increase in plasma ion energy -rr = 2t, (R^f • 23)
~ 2
dw lH\ „
T^~kj'"^'- (16)
In actuality, the energy is absorbed by resonance ions with V|| «
ss "^res ~ T~' These ions may be regarded as magnetic dipoles with the mag-
netic moment _ m,"i ^ Zl which moves in the wave magnetic field with a
38
strength which slowly changes In time and space. Within the frame of
reference in which the wave is at rest, the equation of dipole motion
ma==-V.^Il (17)
dz
leads to the law of energy conservation
+ V-Hz = const. (-L3)
0)
Ions with velocities V|| in the rr-^r - Av < V|| < r™ + Av range, where
Ay
/^-
(19)
are trapped in the potential well and effectively interact with the wave.
It is apparent that the number of resonance particles per unit volvmie is
" ".Vl, (^o*
Due to the shift of ions under the influence of the force -y —,
dz
the distribution function in the resonance region is distorted during the
time T^^^^^^ , during which a trapped particle covers a distance on the
order of the potential well width, i.e.,
^ei-~T7Air- (21)
Relaxation of the distribution function due to collisions in a narrow re-
gion of the width '\^ A v close to v|| = r— - 'vv-j^ occurs during the time period
^.er'-dT- (^^>
where
"^l = .^7S^
4Y2Knae*A'
/45
It is apparent that the ion distribution function will be distorted by an
insignificant amount if Tnonlin. ^^ ""^rel* ^^ critical value of the varia-
ble magnetic field strength H = Hj,^., at which the distribution function dis-
tortion becomes significant, is determined from the condition Tj^q^j]^ ^^^^ _ '\.
-v Trel. Taking into account (19), (21), and (22), we obtain (Ref. 23)
39
Expression (23) for H^^ can be obtained by another method. An in-
crease in the perpendicular velocity component (or a change in the mag-
netic moment) in the presence of a variable magnetic field is determined
in the drift approximation from the following equation
Vj^o> dH,
"- = 2^^ • &-• (24)
The time of nonlinear distortion of the distribution function with
respect to the velocity vj_ under the influence of the field Ea — i.e.,
under the influence of the force "^ — 5. in (24) — is determined according
9z
to the relationship
Tnonlln - -j^^^' (25)
where
(26)
If the ions obtain a velocity increase % Avj_, then the relaxation /46
time of the velocity distribution function is determined by
^r*X -'.(^) • (")
The critical value of the magnetic field strength H = H^j-, at which
great distortion of the ion distribution function f(v||, vj_) occurs close to
V|| = T—, which is determined from the condition i^nonlin '^^rel' coincides
with the value of (23) in the case of :; — 'vv^ . It is also apparent that the
k|| 1
distortion of the velocity distribution function of the particles close to
V|| = T— is not significant, if the energy acquired by resonance particles
■^ II
having the velocity V|| (in the ~ - Av_|_ < v|| < '^ + Avj_ interval), during
K-ll II
the time between two collisions t^^
dw
df
^ ^'-^dr-STT^'-lTT:) ""Tixi, (28)
ires -L
is small as compared with their thermal energy T^. The critical field
strength H, determined from condition — t^^ '^''^1' coincides with (23),
"'- res
40
During heating with fields H = aH^j. under the condition r— o-vi (it is
apparent that the fulfillment of this condition requires that the fre-
quency be increased proportionally to / T^ as the temperature increases)
^ _ "'^^ = °'^o (Toy- (29)
dt T,(<oT.)'/. To(a>oT„)V.\^r,.j •
where Tq , tq and wq are the initial values of Tg, t^ and to.
Assuming that w ~ T^ and integrating equation (29), we obtain the
following expression for the ion temperature (Ref. 23)
'•'-•['+^Ti^r-4'-T^.r-
Plasma heating by means of ion Cherenkov resonance may be intensified, /47
if — instead of one wave — several waves are employed with phase veloci-
ties differing by 2 - 3 Av, so that all particles in the -v^^ < v|| < v^
range may be resonance particles. In order to do this, it is sufficient to
produce a wide wave packet with a phase velocity scattering of A(r~) ~ v^^.
It is apparent that the total number of such waves is 7~'~1/ — ^. It is also
evident that in this case the critical strength of the variable magnetic
field responsible for great distortion of the ion distribution function over
the entire |v| < v^ range is
^cr.tot.~Ai;^cr~l^^cr^o /o^x.)'/. " (31)
In the case of heating by the field H^ ~ "^cr tot ^^ ~ °'^^cr^ ^"^ ^^^
case of a wide wave packet, we have
dw "l (h\ ^ «'^i
-7/-~A5\F„j "'^■- ^- (32)
Integrating expression (22) , we obtain
Formulas (32) and (33) , which determine the ion temperature during the
heating of plasma ions under optimum conditions, when practically all the
plasma ions participate in absorption of the high frequency field energy,
are similar to formulas (1) , (4) , and (7) , which determine Joule electron
41
»■
:*
barely changes). One advantage of this method
heating by subcritical "constant" fields.
In the case of a = 0.01, no 'v lO^^ i/cm^, Tq = 100 ev, Hq % lO^G,
and wq '^ 10^ sec"-^, we obtain: tq o- 10~^ sec, T^ = 50 kev during the time
t '\. 1 sec, while in the case of t = H^ -vSOOG, Ej^ '\j 50 v/cm at the end
of heating oj 'v 20 uq '\. 2-10^ sec"-^ '\' 0.2a)i and H^ % 20G, E^ 'h 40 v/cm
(E^>^ /I^y/e E^
is the fact that energy is transmitted directly to the plasma ion com-
ponent.
In the case of a very non-isothermal plasma (Tg >> T^) , as was shown
in (Ref. 24), a resonance relationship between the outer circuit and the /48
plasma can exist in the case of V . « Vg. In this case for Tg < 10T-j_ ion
absorption is still significant (on the order of electron absorption), and
resonance Vj, ss Vg can exist. The energy absorbed at the maximum increases
by 10 - 100 times as compared with (16) (Ref. 23).
An expression for the high frequency power absorbed by a nonuniform
plasma cylinder in the case of -:-— 'V/ v^ was obtained in (Ref. 25).
As of the present , the effect of Cherenkov absorption of a magneto-
sound wave by plasma ions has not been studied experimentally.
Cherenkov Electron Heating By the Field of a
Magnetosound Wave (Helicons)
A rapid magnetosound wave can be propagated in a plasma with a large
density {^l^^^^e , where Q^ — i / l!!f!^ — Langmuir frequency, Wg, = 2. —
electron gyrofrequency) ; the refractive index of this wave in the a)i<<u)<<a)g
frequency region is determined by the expression
(in this frequency region, rapid magnetosound waves are called "whistling
atmospherics", or simply "atmospherics", "whistles", and "spiral waves") .
Cherenkov whistle absorption by plasma electrons is weak not only
for -r— » Vg, when the damping coefficient is exponentially small (Ref . 26,
■^11
42
27), but also for ^ < v^. In the last case (Ref. 28, 29), we have
k^^v,
(35)
Since the whistle field readily penetrates a dense plasma, these
whistles may be employed to heat the plasma electron component, while
electrons located within the plasma cylinder will be heated up due to /49
the comparatively small absorption coefficient (35) .
If k|| a '\' 1, then — substituting (35) in equation (14) — we obtain
^~(^W„j"'^4;7--^j • (36)
Since the factor — • 2~ differs very little from unity in order
of magnitude, this formula coincides with formula (16) (substituting Tg
by T^) . However, since u >> co-j^ in the case under consideration, and since
formula (16) was obtained for the frequencies u << co- , for the same values
of H electron heating by the whistle field takes place much more rapidly
than Cherenkov heating of ions by a low frequency field.
The change in the electron distribution function with respect to vj_
by the quantity
Ao
-~^^1/ ^
*ii", (37)
is insignificant, if the time of nonlinear distortion of the distribution
function Tj^q^^j^^j^^ r^ — — - — is a little greater than the relaxation time
/Avj_\ ^
T]-el '^''^el J • '^^ critical strength of a variable magnetic field
H = Hj,j., at which the collisions cannot equalize the distribution function,
is determined from the condition T^^onlin '^ ^rel*
^^ (-.)'n"'j '~~(^'^'~w) ■ (^^)
This expression for H^r is obtained under the condition that resonance
electrons ^^ — Ay^<i;|, <-ii-+Au^j, which acquire the following energy
43
per unit of time
dw
dt
res
dw f, (h Y' _, U, STtrtoT-A*''
dt--^'-\Tu] '"^'l^'"^ ' (39)
If H = H during the time "^ t^, acquire energy which is on the order of /50
the thermal energy.
During electron heating by whistle fields with H = cxHj,j., the energy
acquired by one electron on the average per unit of time is
dw '^'T, U Jjlj\''
dt T^(<ox^)'/'\<o/ 87:«„rJ • (40)
Assuming, for purposes of simplicity, that (— . - — %-■] — 1, we obtain
Under optimum conditions, when a wide wave packet is employed and when
a large portion of electrons with a velocity of |v||| < v„ are resonance
"^ r'"'
electrons, electron heating by a field with H2 '^' aHj,^.^ ^^^'x, a v Hj,j.Ho is
determined by the following expression
dw a^r, ^ ^ / a'AV.
These formulas describe Cherenkov electron heating by a whistle field under
optimiffli conditions, if all the plasma electrons participate in energy absorp-
tion. They are similar to formulas (4) and (7) , which may be used to deter-
mine the electron temperature increase during ohmic heating by fields with
subcritical strength.
The study (Ref. 30) was devoted to a theoretical investigation of
Cherenkov electron heating by the electric field of a rapid magnetosoxmd
wave propagated in a nonuniform plasma cylinder. Cherenkov whistle ab-
sorption was determined experimentally in (Ref. 31).
Ion Cyclotron Resonance .
Absorption of Alfygp Wave
Ions are heated most effectively by the field of a high frequency wave
under conditions of ion cyclotron resonance. If the frequency of a wave
44
propagated in a low-pressure plasma is close to the ion cyclotron fre-
quency, then there will be a large number of particles having the velocity
V||, which is close to the particle resonance velocity /51
v^^s=
res- A J . (43)
and effectively Interacting with the wave field. Therefore, the wave
energy absorption by ions will be large. Cyclotron damping of waves is
the inverse of cyclotron radiation of waves by charged particles in a
magnetic field. This effect was first studied in (Ref. 32), where they
investigated the damping of magnetohydrodynamic waves propagated along the
magnetic field. A study of the cyclotron absorption of waves having a
frequency of u) w (u^^ was pursued in (Ref. 33) [see also the studies (Ref. 20,
34, 35)].
As is well known, in the frequency region on the order of ojj^ in a
cold plasma (T = 0) there are two branches of oscillations corresponding
to an Alfven wave (which can only be propagated in the case of lo < o)^)
and corresponding to a rapid magnetosound wave. As the Alfven wave fre-
quency approaches Wj^, its refractive index and cyclotron damping coefficient
increase. In the case of w^ - u < k|| v^, the propagation of this wave by
strong damping is impossible: Re k ~ Im k ~ 1/6^, where 6^ is the depth to
which the field penetrates the plasma (Ref. 28, 36, 37)
A magnetosound wave is absorbed slightly in the case of |w — oj. | < k|| v..
Let us first examine cyclotron heating of a plasma cylinder by an
Alfven wave which is strongly damped. The azimuthal currents (12) excite
this wave if k|| 6^^ ~ 1. Also assiming that k|| a ~ 1 and w^ _ co < k|| v^,
we obtained the following expression from formula (13) for the mean energy
acquired by one ion per unit of time
dr~8^'~\^/'"^/8^n„r,- (45)
It thus follows that the electromagnetic field energy which is acctraiulated
■q— I is absorbed during a period of time on the order of
1
In addition, a comparison of (45) and (16) shows that, for the same /52
variable magnetic field strength, the energy absorbed by the plasma in
the case of co ss Wj^ is considerably greater than in the case of Cherenkov
45
resonance. In the first place, this is due to the difference in the fre-
quencies (■v^)~ <JJ and, in the second place, it is due to a large factor
H§
in expression (32). On the other hand, for the same values of H
SirnoT^
and k|| the electric field strength in the case of cyclotron resonance is
times greater than in the case of Cherenkov resonance (w^^^^gj. —
"^Cher.
the wave frequency under Cherenkov resonance conditions) .
In order to determine the number of resonance ions responsible for
energy absorption during cyclotron resonance, let us Investigate the parti-
cle motion in the field of a flat cyclotron wave :
(46)
Hx = Hsm{kiZ — wt), Hy = —Hcos(kiZ — iot);
H H
Ex = ——cos{ktZ — u)/). Ey = — —sin {kiZ — wt),
"i "«
where n\\ = — = ■ — is the refractive index.
In the absence of a wave, the ions move along a spiral:
V = Vo = {v°^ cos (u),/ + cpo), — v°^ sin (co,-/ + cpo). ^ i )•
When there is a weak field, the velocity perturbation is determined
according to the equations of motion
"x= — J~ cos [k,Z — wt^-\- WiVy-,
K = — ^nr- sin (kiZ — wt\ — WiV^;
z - -^ (vlHy - vlH^ = ^ cos O,
where O = A n 2 + (u)i — u)) ^ + <Po.
Assuming that u = v + iv„, from the first two equations we obtain /53
X J
u 4- iwiu = -e \ ' '.
'«<•" II
Thus, assuming that u = uoe~-^"i , we obtain
46
The third equation can be written as follows
» eu°Hki,
<5 + i,^ " COS = 0.
<^A„ ^ _ (47)
Let us first investigate nonlinear distortion of the distribution func-
tion close to v|| = V , caused by a variable magnetic field. The law of
energy conservation follows from (47)
i. <I)2 + _t 1 sin <D == const.
The time of nonlinear distortion of the distribution function close to
V|| = Vj-gg equals the potential well flight time of trapped particles
having the velocity . ^ ^^, ,, _ „
" ' mkuc
Tnonlin
fe„AU|
We obtain the following expression from the condition t^q^-^^j^j, ^ ^rel ~
for the critical value of the magnetic field strength
^"^ {•^^iY'A Hi ) • (48)
/ Avii \:
This expression may also be obtained from the formula for the nonlinear
decrement of cyclotron wave damping, which is determined on the basis
of the quaslllnear theory (Ref . 38) in the case of v^^^ >> v^, if we set
V
res
Vj^ and k|| ~ 7" in formula (25) of the study (Ref. 38)
However, the influence of the accelerating field E on the distribu- /54
tion function change in a plane perpendicular to Hg is considerably
stronger than the influence of a variable magnetic field on the distribu-
tion function change with respect to velocity along Hq .
Let us determine the time of nonlinear distortion of the distribution
function by the field E:
1
where
^nonlin -- ^ , . ,
47
^"^~°'/|-575;~"'/l(s4)"' <'''
Equating t , . and t i '^ ""^ • I — —I ^^ the case of H -v H^,^, we obtain
1 \Vi /
"cr (a>T.)V.^ //2 j • (50)
We obtain the same expression by assuming that only a group of resonance
particles with v - Av_|_ < V|| < v^.^^ + Avj_ participates in energy absorp-
tion. Resonance particles acquire the following energy per unit of time
dw
IF
tes V^j '"'^'(^J •
In the case of H '\. H^,^, during the time ^ t^^ these particles collect
the energy '^ T^. The critical field strength (50) is considerably less
than (48) . This means that the influence of nonlinear effects caused by
the electric field is manifested for smaller field strengths than is the
case for the influence of a variable magnetic field.
^^ dw
In the case of H = aH^^., we obtain the following expression for -r—
dt ^.r,y/.\8nn,T, ,^ („,^)'A I 87.«„ro j [Ti j ' (5^)
Thus, we have
Ti-T,
,+,_■■' f"o
"■^0 {"^o)'''' \»^'hTi
(52)
Let us now present a numerical example. Let us set uq 'v 10 ^ cm~^, /55
Ho ^ 5'103g, uij_ '^ 5*10'^ sec-^ and a -v 1. Then in the case of T^^ = 10 ev,
we obtain: t^ -v. 3'10"^ sec, H '\. 2G and -^ "^ 300 kev/sec; if T^ 'V' 100 ev,
then T^ '^10"'* sec, E'^ 0.4G and ^ '^ 30 kev/sec; if T^ 'x, 1 kev, then
T4 '\j 3*10"^ sec, H 'v O.IG and -r-r ^ 3 kev/sec.
-*- at
In the case of Tq "^ 100 ev and H ^ H , the temperature T. ^ 10 kev
is achieved during the time t '\/ 3 sec.
48
Cyclotron heating by fields with subcritical strengths may be in-
tensified, if several waves (wave packet) are employed having phase
velocities which differ with respect to 2 - 3 AvJ_. The number of such
waves IS '^ —i. , and the critical strength of the total magnetic field is
AvJ_
^cr. tot "^ ^ Hpr^O' where B.^^ is determined according to (50). In the
dw
case of heating by the field H '^' otH tot' ^^ this case for -rr we ob-
tain (32), and for T^ we obtain (33).
Cyclotron wave absorption in a plasma cylinder with a constant
(over the cross section) plane and temperature was analyzed in (Ref . 33,
39). The absorption of long wave oscillations in a nonuniform plasma
cylinder was studied in (Ref. 40).
In the case of a plasma with a large density and magnetic fields
with a high strength, the skin depth S^ (44) is small. When 6^ is less
than the plasma radius, plasma heating by an Alfven wave is ineffective,
since the wave energy is absorbed only by ions located on the periphery
of the plasma cylinder.
In order to avoid this difficulty, Stlx (Ref. 41) employed the in-
genious idea of "magnetic beaches". If the plasma cylinder is placed in
the field Hq with a slowly decreasing strength, then cyclotron damping is
exponentially small in the region w^ - w >> k|| v^^. An Alfven wave readily
penetrates the plasma, and they may be excited resonantly. When the wave
is propagated along Hg , the difference w. - oi decreases, while the refrac-
tive index n|| '\- 1/ u. - to and the damping coefficient x— exp — o "o - /56
^ [ 2* II t;f J
increase. If the magnetic field decreases smoothly, the reflection co-
efficient will be small, and the wave will be absorbed at the approach
to the region of strong cyclotron damping, where w^^ - u^ k|| v- (region
of the "magnetic beach"). The field behavior close to the magnetic beach
was determined in (Ref. 42, 43). Absorption of Alfven waves at the magnetic
beaches was determined experimentally in (Ref. 44). Experiments perfoinned
at Princeton closely coincide with the theoretical computations of cyclotron
damping in a linear approximation [see the simmiary in (Ref. 45)]. On the
other hand, in several devices for ion cyclotron heating (see, for example,
[Ref. 46]) the variable field strengths are on the order of, or even con-
siderably larger than, the critical strengths. Under these conditions,
we must expect a decrease in the field energy absorption, as compared with
the case of weak fields.
49
Cyclotron Absorptio n of a Magnetosoun d Wav e
The above-mentioned difficulty entailed in high frequencv heating
of a plasma having a large density by a strongly damped Alfven wave is
unimportant for a rapid magnetosound wave. A magnetosound wave is ab-
sorbed in the case of o) « u^ to a considerably lesser extent, and can
penetrate the plasma readily. The damping coefficinet of this wave in
the case of [ai - (^j| ^ k]| v^^ is on the order of (Ref. 28)
(53)
c
where nj_ '^ ~.
^A
If k|| a '^' 1, we obtain the following expression (Ref. 39) for the
energy absorbed on the average by one ion per unit of time from the formula
(13) , taking into account (53)
Plasma heating by a magnetosound wave is greatly intensified, if the
frequency to ;i; u^ coincides with the eigen oscillation frequency of a plasma
cylinder w^^g. In this case, the field strength in the plasma increases /57
in the case of loj_„„ - ail :^o)i/ ""° '" , as compared with the non-
res y ^2
resonance case, by a factor of 1 / ^o *
SmiaTi '
It is apparent that we then have
(56)
Multiple resonance: u) = 2w£, may be employed to heat a plasma with a
large density by a magnetosound wave. The damping coefficient of the mag-
netosound wave in the case of |ci) - Zw^ | ^ k|| v-j^ is (Ref. 28)
^_ /-8^/_Q^Y (57)
«x V "" Vw'J
where 2- = i /'^'^^""q is the ion Langmuir frequency. It follows
y '"i
50
from (57) that in a plasma having a large density Qj_ > k|| c resonance
at the multiple frequency w « 2a)j is more advantageous than at the main
frequency .
The heating of a plasma cylinder which is uniform across its cross
section was examined for o) « nto^^ (n = 2,3, ...) in (Ref. 47), and the
case of wavelengths (k 1 1 a << l)in a nonuniform plasma was investigated in
(Ref. 48). Cyclotron absorption of magnetosound waves in the case of
0) = 03^ and to = 2co. has not been determined experimentally as yet, although
the inverse effect — cyclotron radiation of ions in a dense plasma —
was studied recently (Ref. 49).
The statements presented above Illustrate the following:
(1) Cherenkov and cyclotron plasma heating with weak fields, when
distortion of the ion distribution function is compensated by collisions,
occur at the same rate as ohmic heating by a "constant" electric field, /58
whose strength is less than the critical strength;
(2) Plasma heating by weak fields up to thermonuclear temperatures
(T^ ~ 50 kev) takes place over a long period of time. However, this time
is less than the time for containing a plasma in a thermonuclear reactor
with a positive balance;
(3) When heating is performed with fields having subcritlcal strengths,
it is more advantageous to employ a plasma with a great density, since col-
lisions occur more frequently in it, the critical field strengths are larger,
and the heating time is less than (t ~ — ) .
no
REFERENCES
1. Transactions of the Second International Conference on the Peaceful
Utilization of Atomic Energy. Reports of Soviet Scientists (Trudy II
mezhdunarodnoy konferentsii po miromu ispol'zovaniyu atomnoy energii.
Doklady sovetskikh uchenykh) . Gosatomizdat , Moscow, 1959.
2. Transactions of the Second International Conference on the Peaceful
Utilization of Atomic Energy. Selected Reports of Foreign Scientists.
Physics of a Hot Plasma and Thermonuclear Reactions (Trudy II mezhdun-
arodnoy konferentsii po mirnomu ispol'zovaniyu atomnoy energii.
Izbrannyye doklady inostrannykh uchenykh. Fizika goryachey plazmy
i termoyadernyye reaktsii) . Gosatomizdat, Moscow, 1959.
3. Zavoyskiy, Ye. K. Atomnaya Energiya, 14, 57, 1963.
4. Fanchenko, S. D. , Demidov, B. A., Yelagin, N. I., Ryutov, D. D. Zhurnal
51
Eksperimental'noy i Teoreticheskoy Fizlki, 46, 497, 1964.
5. Chechkin, V. V., Vasil'yev, M. P., Grigor'yeva, L. I., Smerdov, B. I.
Present Collection, 7.
6. Sukhomlin, Ye. A., Suprunenko, V. A., Reva, N. I., Tolok, V. T.
Present Collection, 118.
7. Alexeff, B. Phys. Rev. Lett., 7, 196, 1963.
8. Spittser, L. Physics of a Fully Ionized Plasma. XL, Moscow, 1957.
9. Buneman, 0. Phys. Rev. Lett., 1, 104, 1958; Phys. Rev., 115, 503,
1959.
10. Suprunenko, V. A. Author's Abstract of Candidate's Dissertation
(Avtoreferat kandidatskoy dissertatsii) , 1963.
11. Suprunenko, V. A., Faynberg, Ya. B. , Tolok, V. T., Sukhomlin, Ye. A.,
Volkov, Ye. D. , Rudnev, N. I., Reva, N. I. Atomnaya Energiya, 14,
349, 1963.
12. Gordeyev, G. V. Zhurnal Eksperimental'noy i Teoreticheskoy Fiziki,
27, 19, 1954.
13. Drummond, J. E., Rosenbluth, M. N. Phys. Fl. , 5, 1507, 1962.
14. Vedenov, A. A., Velikhov, Ye. P., Sagdeyev, R. Z. Yadernyy Sintez,
1, 168, 1961.
15. Vedenov, A. A. Atomnaya Energiya, 12, 2, 1962.
16. Petviashvili, V. I. Author's Abstract of Candidate's Dissertation
(Avtoreferat kandidatskoy dissertatsii), Tbilisi, 1963.
17. Kadomtsev, B. B. , Petviashvili, V. I. Zhurnal Eksperimental'noy i
Teoreticheskoy Fiziki, 42, 2234, 1962.
18. Petviashvili, V. I. Zhurnal Eksperimental'noy i Teoreticheskoy
Fiziki, 45, 1427, 1963.
19. Lowson, J. D. Proc. Phys. Soc. , 70B, 6, 1957; in the Book: Controlled
Thermonuclear Reactions (Upravlyayemyye termoyadern3rye reaktsii) .
Gosatomizdat, Moscow, 12, 1960.
20. Berger, J. M. , Newcomb, W. A., Dawson, J. M. , Frieman, E. A., Kulszud,
R. M. , Lenard, A. in the Book: Transactions of the Second Inter-
national Conference on the Peaceful Utilization of Atomic Energy.
Selected Reports of Foreign Scientists. Physics of a Hot Plasma
52
and Thermonuclear Reactions (Trudy II mezhdunarodnoy konferentsii /59
po mirnomu ispol'zovaniyu atomnoy energll. Izbrannyye doklady ino-
strannykh uchenykh, Fizika goryachey plazmy i termoyadernyye reakt-
sii). Gosatomlzdat, Moscow, 181, 1959.
21. Sllln, V. P. Zhurnal Eksperimental'noy i Teoretlcheskoy Fizlkl, 23,
649, 1952.
22. Stepanov, K. N. Zhurnal Eksperimental'noy i Teoretlcheskoy Fizikl,
34, 1292, 1958.
23. Stepanov, K. N. Zhurnal Eksperimental'noy i Teoretlcheskoy Fizikl,
45, 1196, 1963.
24. Shapiro, V. D. In the Book: Plasma Physics and Problems of Controlled
Thermonuclear Synthesis (Fizikl plazmy i problemy upravlyayemogo
termoyadernogo slnteza) , 1. Izdatel'stvo AN USSR, Kiev, 62, 1962.
25. Dolgopolov, V. v., Stepanov, K. N. Yadernyy Sintez, 3, 205, 1963.
26. Shafranov, V. D. Zhurnal Eksperimental'noy i Teoretlcheskoy Fizikl,
34, 1475, 1958.
27. Gershman, B. N. Izvestlya Vuzov. Radiofizika, 1, 4-5, 49, 1958.
28. Stepanov, K. N. Zhurnal Eksperimental'noy i Teoretlcheskoy Fiziki,
38, 265, 1960.
29. Stepanov, K. N. Izvestlya Vuzov. Radiofizika, 6, 403, 1963.
30. Dolgopolov, V. v., Yermakov, A. I., Nazarov, N. I., Stepanov, K. N. ,
Tolok, V. T. Yadernyy Sintez, 3, 251, 1963,
31. Nazarov, N. I., Yermakov, A. I., Dolgopolov, V. V., Stepanov, K. N. ,
Tolok, V. T. Yadernyy Sintez, 3, 255, 1963; Zhurnal Eksperimental'-
noy i Teoretlcheskoy Fiziki, 45, 1260, 1963.
32. Gershman, B. N. Zhurnal Eksperimental'noy 1 Teoretlcheskoy Fiziki,
24, 453, 1953.
33. Stix, T. H. Phys. Fl. , 1, 308, 1958.
34. Shafranov, V. D. In the Book: Plasma Physics and Problems of Con-
trolled Thermonuclear Reactions (Fizika plazmy i problemy upravlya-
yemykh termoyadernykh reaktsiy) , 4. Izdatel'stvo AN SSSR, Moscow,
426, 1958.
35. Sagdeyev, R. Z. , Shafranov, V. D. In the Book: Plasma Physics and
Problems of Controlled Thermonuclear Reactions (Fizika plazmy 1
53
problemy upravlyayemykh termoyadernykh reaktsiy) , 4. Izdatel'stvo
AN SSSR, Moscow, 430, 1958.
36. Stepanov, K. N. Zhurnal Eksperlmental'noy 1 Teoreticheskoy Fiziki,
36, 1457, 1959.
37. Doyle, P. H. , Neufeld, J. Phys. Fl. , 2, 39, 1959.
38. Yakimenko, V. L. Zhurnal Eksperlmental'noy i Teoreticheskoy Fiziki,
44, 1534, 1963.
39. Vasil'yev, M. P., Dolgopolov, V. V., Grigor'yeva, L. I., Smerdov,
B. I., Stepanov, K. N. , Chechkin, V. V. In the Book: Plasma
Physics and Problems of Controlled Thermonuclear Synthesis
(Fizika plazmy i problemy upravlyayemogo termoyadernogo sinteza) ,
3. Izdatel'stvo AN USSR, Kiev, 96, 1963; Zhurnal Teoreticheskoy
Fiziki, 34, 974, 1964.
40. Dolgopolov, V. v., Stepanov, K. N. Zhurnal Teoreticheskoy Fiziki,
34, No. 7, 1964.
41. Stix, T. H. Phys. Fl. , 3, 19, 1960.
42. Stepanov, K. N. Yademyy Sintez, 3, 154, 1963.
43. Piliya, A. D. Zhurnal Teoreticheskoy Fiziki, 34, 93, 1964.
44. Hook, W. H. , Tenney, F. H. , Brennan, M. H., Hill, H. M. , Stix, T. H.
Phys. Fl., 4, 1131, 1961.
45. Hook, W. H. Nucl. Fusion, 4, 33, 1964.
46. Nazarov, N. I., Yermakov, A. I., Tolok, V. T. Present Collection, 1.
47. Dolgopolov, V. v., Stepanov, K. N. Zhurnal Teoreticheskoy Fiziki,
33, 1196, 1964.
48. Dolgopolov, V. v., Stepanov, K. N. In the Book: Plasma Physics and
Problems of Controlled Thermonuclear Synthesis (Fizika plazmy i prob-
lemy upravlyayemogo termoyadernogo sinteza), 4. Izdatel'stvo AN
USSR, Kiev, 38, 1965.
49. Golovin, I. N. Artemenkov, L. I., Bogdanov, G. F. , Panov, D. A.,
Plstunovich, V. I., Semashko, N. N. Uspekhi Fizicheskikh Nauk,
73, 685, 1961.
54
DIELECTRIC CONSTANT OF A PLASMA. IN A DIRECT PINCH MAGNETIC
FIELD AND IN A DIRECT HELICAL MAGNETIC FIELD
V. F. Aleksin, V. I. Yashin
Di electric Constant of a Plasma in a /60
Direct Pinch Magnetic Field
The study of electromagnetic oscillations in a nonuniform plasma
has aroused a great deal of interest recently. This is primarily due to
the discovery of several instabilities. A great ntomber of these studies*
has been devoted to studying the electromagnetic properties of a slightly-
nonuniform plasma located either in an almost uniform magnetic field with
parallel force lines, or in an axially symmetrical magnetic field with
helical magnetic force lines.
In conjunction with these cases, when investigating the problem of
controlled thermonuclear synthesis it is very important to study the plasma
electromagnetic oscillations in pinch magnetic fields and in helical mag-
netic fields and in a stellarator with helical current winding (Ref. 3),
which has a more complex structure than magnetic force lines.
As is well known, the electromagnetic properties of media are de-
scribed by the dielectric constant tensor or the electroconductivity tensor
related to it. In slightly nonuniform media, one can introduce quantities
which are similar to the electroconductivity tensor and the dielectric
constant tensor. In contrast to a uniform plasma, these tensors depend on
spatial variables in wave vector space k and the frequency w.
If E(k,(o) is the electric field strength, then the density of the
current induced by this field can be written as follows
ia{r, = Jc?kdu)o„p(k, w, r)£"p(k, w)e' i^'-""), (1)
where a o(k, w, r) is the electroconductivity tensor of a nonuniform plasma,
which is related to the dielectric constant tensor by the well-known rela- /61
tionship
Sap(k, 0), r) = 8ap + -;^a«p(k, o>, r). (2)
The explicit form of the tensor o^^gCk, (o, r) may be found by different
A detailed list of the literature is presented in the review articles
of A. B. Mikhaylovskiy (Ref. 1), A. A. Rukhadze, and V. P. Silin (Ref. 2)
55
methods. In our case, it is advantageous to employ the method advanced by
VJD. Shaf ranov (Ref. 4). In order to do this, it is necessary to determine
the trajectory of the unperturbed particle motion and to solve the kinetic
equation by the characteristics method.
In the absence of a balanced electric field, the unperturbed charged
particle motion may be described by the equations
S=i™; §-v. (3)
where B is the magnetic field strength in the plasma; e and m — particle
charge and mass, respectively. In the cylindrical coordinate system, the
strength components of a nonvortical, direct pinch magnetic field have the
following form
Br = Bo ^ §n sin az«;
Bz = bJi+ J^fnCOsanzY, (4)
where Bq is a uniform magnetic field; a = - — ; £„(!■) and g^C^) — functions
of the coordinate r which are connected by the relationships
"y^ no.gr,- i^=rno.U. (5)
In the presence of a plasma, the field components, which are related to
the pressure gradient and the longitudinal current and which may be found
from the equations of plasma equilibrium, must be added to the nonvortical
field (4).
A solution of equations (3) -i-n the general case of arbitrary fields /62
entails considerable difficulties. Let us examine the case (which is
(a=U^«i)
of practical interest) of a low-pressure plasma (^ = ^^z "^"^ 1) without a
longitudinal current, in a pinch magnetic field (4) having a large uniform
component Bo(fj^~ gn~ ^ << 1) •
In solving nonlinear equations (3) , the presence of a small parameter
6 makes it possible to employ — along with the drift approximation — the
method of averaging (Ref. 5) when solving drift equations for flying parti-
cles. The number of particles which are blocked is small, and their contri-
bution to the tensor a^o may be disregarded. Avoiding cumbersome computa-
tions, let us derive the final result of solving the equations of motion (3)
within an accuracy of terms on the order of — ( *^r ~ ^"^ ) inclusively
a)g
y B mc y
56
r{t) =r — Yi ^[cosnci.{v ^t + z)—cosna.z]~
~;;r:t^'"(T — "'sO — sinTl;
? W = ? + ^ + 2j^^^77i;^S-«[sin ««(«„/ + z) _
n=l
— Sin naz] + ^— - [cos (y — wbO — cos 7];
00
- ^ 2 [sin na (w n if + z) — sin naz];
2nav I,
" U (^) "" " U ~ Zj 2^ /n [^°s «a (0 ^ + z) — COS naz];
n=l
v±{t) = fx +2j 2""-L/"I^os«a(y / + z) —cos naz].
(6)
n=l
where v|| and v_[_ are the velocity components which are longitudinal and /^3
transverse to the field, respectively; y — the initial phase of particle
rotation around the center of a Larmor circle; v"^ — the averaged velocity
component related to the drift of the Larmor circle center:
(7)
ro>gu2
For purposes of simplicity, we have omitted the index for quantities
taken at the initial moment of time t = in formulas (6) and (7). We
shall also do this from this point on.
By solving the kinetic equation, with no allowance for close collisions,
by the method of characteristics, we can find the expression for the electro-
conductivity tensor in a cylindrical coordinate system from the equation
for the induced current density
57
I I IMI nil ^■■^IHI^i^HHaHII^H^^^BH^H^^^I^I^^H^^HmmHinillllll ■■■■■■■■■
where Fg is the equilibrium distribution function of the particles; k —
m
the wave vector with the components k^, k^, kj, (kj, = — , m — whole numbers);
E — the sum with respect to ions and electrons. The electric field may
i,e
■>•
be determined by the expression
E (r, U)) = S I dkrdk, E {K, k., m) d'^rr^rik.'^tm, _ ^^ ^
For a slightly nonuniform plasma in a strong magnetic field, we may
write
^0 = (i + ^y^ ^^^)f{v„ ... w), (10) ^^
for the equilibrium distribution function within an accuracy of terms on
vp
the first order of smallness with respect to the parameter (vp —
thermal particle velocity, a — characteristic dimension of the nonuni-
form! ty for the main plasma state). Here, x — unit vector in the field
direction; f — arbitrary function of the velocities v||, vj_ and of the
variable ¥ which is an integral of the drift equations. Within an accuracy
of the terms — , the quantity W coincides with the integral of the force
line equations
^=./-^ + 2r2|cos«az. (H)
Due to the smallness of the parameter 6, we may approximately compute
the function f which depends only on the coordinate r. We should note that,in
order to avoid this, in a plasma with the selected distribution function
div j =i^ we may add the component vB x 1 ^5^ " "^^^^ ^° ^^^ function Fq ,
where Integration is performed along the magnetic force line. However, as
may be readily seen, in our approximation this term is small.
Let us select the Maxwell distribution of particles with a nonuniform
equilibrium density ng and the temperature T, which depend on 'P, as f :
, = „.(,-fJ%(-^i^).
(12)
Retaining the important terms in (8) , after simple transformations we ob-
tain
i.c ^ ' (13)
58
where the following notation is employed:
1=1 s, s'
^n = arctg —
(14) /65
2jr
^"P = —2!^ J ^T^a J yp (0 ^ (0 d^;
^ (/) = exp {— / (o) — tiasv i—k^v^ — k^v,, )t--ax
X [sin(-)f — co£/ — (j;)_ sin (y — <}))]};
The velocity components which are included in the expression for the ten-
sor Qjjjg in a cylindrical coordinate system have the following form
00
"r{t) = V :^ COS {-i — iost) + 1] g^v , sitina {v it + z);
r-^'=€'xSm(T-«.BO+ 1 2^^ V .cosna(^.^ + 2) + cr,; (15)
°° 2
^^ (0 = y I — X^ /" COS na (y , / + 2).
By employing them, we can obtain the explicit form of the tensor Q g
where the vectors q and q have the following form
qr = Ux [l^p (?) cos if + iJ'p iS) sin ^\ + Vi~Jp Q.)\
9f = ^x [f -^p (^) sin <J; — iVp iS) cos ijjj + F/p {^);.'
59
gr=^v.[^Jp (S) COS t - iJ'p (^) Sin ^) 4°' - 4 I, , /, (^) S ^/4'V'«; (^7) _£66_
9^ = 0^ [|- /p (S) sin <{. + f-y; (S) cos <i<]4°> + Vp(04°';
cJJ' =: (u) — pcofi -^ nasu I — lav , — fe^u, — k^v j )~'
(J (5) is the Bessel function; the sum with respect to 1 is taken from
-00 to +00, and it is thus assxamed that g_n =-gj,, f. = jj, and f q = 0) .
As is well known, the electroconductivity tensor may be employed
to obtain the expression for the polarizability vector x which charac-
terizes the density of the induced charge p ,
p = y dlcdoje-- C^—O X,E^ (k, w). (18)
The components of the polarizability vector have the following form
Za = ir. 2 e^ J ^y ^ dvlML ^f(v,, v\, W) $] h (^) 9- (19)
I, t p=s — oo
The expressions for the electroconductivity tensor and the polariza-
bility vector may be simplified in the case of electromagnetic oscilla-
tions, whose wavelength is much greater than the pinch modulation depth
(rijj << 1). In this case, we must get rid of complex svmis and products,
since the terms with s = s' =0 and the operator M = 1 will make the
main contribution.
Plasma Dielectric Constant in a Dielectric Helical
Magnetic Field of a Stellarator
The strength components of a nonvortical magnetic field from a helical
current winding with the finite step L have the following form (Ref . 6)
5r=5„ 2^„sin«9; ^^ = ^0 S^/'.cosnO; (20) J67_
n=l
Bg = Bq — o.rB^,
where Bq is the strength of a viniform, longitudinal magnetic field;
2ir
6 = (J) - az; a = ~7~', gn(^) ^'^'^ ^n(''^) — functions of the coordinate r,
which are related by the following relationships
60
Just as in the case of a pinch, field, when deriving the explicit
form of the electroconductivlty tensor, we investigated a plasma with a
small gasokinetic pressure (6 << 1), without a longitudinal current, in
a helical magnetic field having a large axial component
Within an accuracy of terms on the order of — , the particle tra-
jectories and velocities may be described by the following expressions
— ("bO — sin y];
X [COS (y — iOBt) — COS yI;
2 " -
z(O=z+^.^ + 2^i;'J?[sin«(G+^)-sin/z0]; (22)
f/- (0 = y-L cos (y — mst) + Vi YiSnS^mnl^ -{-—A;
/•M2 = y^ sin (y — (BflO + y I 2/« cos n ^G + ^j + y^;
o ~ —
Vz (0 = ^^ — 2^ «"■ 2j /" COS «. [9 + —j.
n=l
where v., v^ and Vg = va - arv^ are the averaged velocity components re- /68
lated to the motion of the center of a Larmor circle along the averaged
force line and drift in a nonuniform field:
(23)
y = artv 1 -1 — Ml + — '- «2 -I ^ "s".
fz = C, - -4 t> II 2j (/" + g") - ^ r'ao.3 "3-
Here x is the torsion angle of the magnetic force lines:
n=l
61
"1 == I S (/« + si - 2nfngn - 3aV^«/„^„);
n
„, = I- j; {fl +gl- 2nfngn - aV^n/„g„) + '-^; (25)
n
«3 = — g- 1] a^r^nfngn-
n
The distribution function Fq in a slightly nonuniform plasma may also
be determined by (10) , in which 'F describes the equation of magnetic sur-
faces:
VF = r2 — 2rS^cosrt0. (26)
n
We again select the fimction f as a Maxwell function, with the density and
temperature dependent on the magnetic surfaces V.
As a result of the computations, we obtained the following from the
general expression for o^^ (8)
where
..^=^i^Y,^'\ML^f{v,,v„^)Y, q^q^dv^dvl, (27)
A A
ML:
I. I ' P=-
M = n 2j ^sW^'(%)e ^'.
n=\ s, s'
<7. = y X [f ^p (^) cos ^ + iV; ($) sin <!>] + J^ y , /p {%);
<7, = t; X [f ^P (^) sin t - fV'p (?) cos t] + ^./p (S) + g-^ " 1 ^p (^);
^, = t; X [f ^p (^) cos i> — i7'p (5) sin <1^] c^ — ^ ^ » /p (^) X
(28) /69
62
^, = f X (f Jp(i) sin <}. + iJ'p (?) cos t) c<,°> + v,Jp (E) 4°> +
The quantities kx, kj_, ij;, g and the operator L are determined by the ex-
pressions (14) ; it is assumed that g_ = -g , -f = f and f q = in the
Jl^ A/ Xi 10
sums over the index Jl . Correspondingly, the plasma polarizability vector
in a helical magnetic field is
A A 1
p=
X^a = w^e^ j ML j7/(o,, Ux, 'P) S Jp{i)qJVidv].. ^29)
Just as in a pinch field, expressions (27) and (29) may be simpli- /70
field, if the oscillation wave length is much greater than the difference
between the maximum and minimum radii of the magnetic surface (n << 1) .
We may employ the expressions obtained for o^o and x„ to study the electro-
magnetic oscillations and plasma stability in pinch magnetic fields and
in helical magnetic fields. This will be the subject of future research.
REFERENCES
1. Mikhaylovskiy , A. B. In the Book: Problems of Plasma Theory (Voprosy
teorii plazmy) , 3. Gosatomizdat , Moscow, 141, 1963.
2. Rukhadze, A. A., Silin, V. P. Uspekhi Fizicheskikh Nauk, 82, 499, 1964.
3. Spitzer, L. In the Book: Physics of a Hot Plasma and Thermonuclear
Reactions, 1. Atomizdat, Moscow, 505, 1959.
4. Shafranov, V. D. In the Book: Problems of Plasma Theory (Voprosy
teorii plazmy), 3. Gosatomizdat, Moscow, 3, 1963.
5. Bogolyubov, N. N. , Mitropol'skiy, Yu. A. Asjnnptotic Methods in the
Theory of Nonlinear Oscillations (Asimptoticheskiye metody v teorii
nelineynykh kolebaniy) . Fizmatgiz, Moscow, 1958.
6. Morozov, A. I., Solov'yev, L. S. Zhumal Teoreticheskoy Fiziki, 30,
271, 1960.
63
SECTION II
LINEAR PLASMAJ)SCILLA TIONS /71
KINETIC THEORY OF ELECTROMAGNETIC WAVES IN A CONFINED PLASMA
A. N. Kondratenko
Problems of electromagnetic wave propagation in a confined plasma
are of considerable interest to studies on methods of plasma heating,
acceleration of charged particles, plasma diagnostics, and other possi-
ble applications. The hydrodynamic theory of plasma wave guides for slow
waves has been studied in great detail [see, for example, the articles
(Ref. 1 - 4)]. However, the hydrodynamic theory does not encompass the
important phenomena related to the particle thermal motion — for example,
wave damping which is particularly great at small phase velocities.
Since the phase velocity of a propagated wave V^ is less in the wave
guides of slow waves than the speed of light, and since it may be compara-
ble to the mean thermal velocity of electrons Vj^ or ions v>pj, the neces-
sity of a kinetic examination becomes readily apparent.
On the other hand, the confinement of a plasma leads to a new type 111
of wave — surface waves — whose damping, as was shown in (Ref. 5, 6),
is proportional to the thermal velocity of plasma electrons for v^jg << V$.
In a nonconfined plasma, where there is no surface wave, the damping of
the longitudinal three-dimensional wave is exponentially small (Ref. 7).
Foirmulation of the Problem
Let us investigate the propagation of slow electromagnetic waves in
a plasma layer which is 2a thick in one direction, and is not confined in
the two other directions. As is known (Ref. 4), waves propagated under
these conditions are surface waves when there is no magnetic field.
A self-consistent system of equations describing these processes con-
sists of the Maxwell equations
™tE==-^f: (1)
-tH=|.f + ^- (2)
and a linearized kinetic equation for the deviation of f^^ from the equill-
briiam distribution function fo„ of the a type of particles (a = i — ions;
a = e — electrons) , in which we shall disregard particle pair collisions
64
t+'J + ^(^ + }l«."l)^--0. (3,
The standard notation is employed in equations (1) - (3) . We shall select
the coordinate axes so that the z axis coincides with the direction of
the wave propagation, and the x axis is perpendicular to the layer. The
YZ plane is the plane of symmetry.
Equations (1) - (3) must be supplemented by the boundary conditions.
Let us assume that f„ = fj + f^, where fj is the distribution function
for v^ > 0, and f~ — for v,, < 0. We shall select the conditions of the
mirror image (Ref. 8) (the final result does not depend quantitatively
on the reflection condition) as the boundary conditions for the function f *
}t{ + a,V:c>0,v,)=fri±a,V;,<0,v,). (4)
We obtain the boundary conditions for the fields from the Maxwell equa- /73
tions (1) and (2) , integrating them over an infinitely thin layer which
encompasses the plasma-vacuum boundary:
Dispersion Equation
We can write the dependence of the distribution function fa on
time and the cordinate z for the fielctein the form exp i(k3Z - cot). If
fOQ( is the function of energy, we obtain the following from equation (3)
with allowance for the boundary conditions
— a
J^^e^n^^A 0^^^ ^ (6)
— a
_/•£.($) 3^ sin T (a -S)|,
where
fejV^— 0>
In order to compute the currents j^, and j^, it is necessary to deter-
mine the simi and the difference ft, + fl^ Employing the values (6), we
obtain
65
a
— a
— a
a
2ie^ a/„, p (8)
— a
where ... ^ , . x ► /74
„ , „ ^ 1 f cos T (S — a) cos 7 (X + a), ;c < ^,
sin27a\(,Qs^(;j._Q,)(,os7(S + a), a:> $; (9)
/( (x ^)^ ^ fsin7(^ — a)sin-r(A: + a), ^ < ^,
""^ • ' sin2-jff\sin^(;c — G)sinf (^+a), a:>5. (10)
The kernels K^ and K2 , which are expanded in Fourier series , have
the following form
'cos o„S cos a„ X
Y' ''^°^ "n^ '^OS "n *
Ki (x, i) = lL —Tir- ' (11)
n=0
_, sin a^S sin a^ x
/C2(^, = ^2j f^-a^""' (12)
where a = — ; the prime over the sum indicates that the sum term corres-
n a -1
ponding to n = must be multiplied by y.
Employing the values (7) , (8) , (11) and (12) and using equation (1)
to change from the fields E^, E^. to the fields E^, Hy, we may obtain the
formulas for determining the currents :
;^ sin a J ". ,.,v2.^f^ }j^^(dh. 7 ^M,
+ 'P J "y sin «"W^ 2j ;^ J "'"^ J f - «r 5^; ' (13)
66
. '^' COS a„ ^ h A 2ie\ f v,dv^ f dv^ f a/o,
n=0 \ a=i — ~
a=/ — oo
where
ckc
The solution of the Maxwell equations (1) and (2) for determining /75
the fields E^ and H^ leads to the following equations
f-/^(l-p^)/f, = l^/.; (15)
We may write the solution of the integro-differential system of
equations (13) - (16) in the following form
EzU) = ^'fan cos a„A:; Hy{x) = '^HynS'manX. (17)
We obtained the following value for the Fourier components E^.^^:
£.„ = _(_-!)« I //,(a)M+iML^\ (18)
where
A= J--/1
(19)
[i~ - A,) [a, + /| (1 - n] + i^B - ««) (^2 + ^n);
0=t — oo
67
I I I II I I II iiiiiiniHiaiM ■■■iini i ■
Here fi^ = " ; pg — equilibritmi ion density which equals the
c m
■"a
equilibrium electron density; f is the density normalized to unity
(/f dV = 1) .
oa
Selecting Maxwell distributions as the equilibrium distribution /76
functions f , we may write the equation for the plasma impedance
" n=o
1-PIi-SSQ'^
(21)
We thus have
i-PV-i]S<3-
a={
V^Q^
a=l"
Qi. = I a. J ^2 (a. - bz) e-^' [(a, - 6z)^ /„ (z) - K^];
Q2a = 1 0. f dzz^ (a, — bz) er^'Ia (z);
Qa. = ?- a,6 J dz2 (a, - fcz)« e-^7a (z)
(22)
(23)
; wt
the mean thermal velocity of a type of particles ;
b=^; h{z)= f t^ )
Equating the plasma impedance (21) to the vacutmi impedance
^z(«) _. c ^
iHJa)- 0. '^s^^
2\2
-ff.
(24)
we obtain the dispersion equation
68
J*3 2ja
i-PMi-LipQ
n=0
s
Q^
!u
= (1 - P«)2
(25)
The sum (25) consists of the real and imaginary parts, since each Q con-
tains the real part which equals the main value, and the imaginary part
which is proportional to the residue of the integral I (z) .
/77
High Frequency Oscillations
Let us set Vrt = 0, — -' <C Ir wi/ ->- oo.
We shall employ the following notation
Pa = ?^' Im
n=0 *- ^
Qi
(26)
(27)
Since a^^ are evenly included in the sum (26) , we may change from the
sum to the integral (Ref . 8)
^^=^^J^[^-p^(i-S'^^('^)
dq.
(28)
in which integration over the contour C is performed from -«> to -H=°, passing
around the singular points q = — from above. Let us deform the integration
3.
contour, and let us enclose it in the upper half-plane. Let us set
qo = ikj_ — the value of q at which A(q) =0. We then have
(29)
since the integrals Qj have the following principal values Oj (j =1, 2, 3)
when the electron pressure is disregarded:
When computing the sum P2, we should note that — since a =
= b T >> b — for small n, just as for a >> 1, the integral residue
ICgV-pg e
I(z) is exponentially small. Therefore, components with n which is
69
larger than a certain ng (i.e., shortwave components of Fourier expan- /78
sion) , which may be determined from the condition o^ "^ 1, make the main
contribution to the sum P2. For such n, b^ << 1, and Qj have the
following values (we shall omit the index e) :
Q^ = 20^ [ -1 + a7»= \2]e''dt - iV^
Q2 = a^°' (2 ] e''dt ~ i yii) ;
(30)
Q,=2ob'
_a + U — i) e-o= (2 ]e''dt - i V'
Since Qi '\' a^, Q2 "^ o, -^ '^ a, in the case of b^ << a^ , b^ << 1 for A,
b
which is determined by equation (22) , we obtain the following expression
A = -4(i-|q,). (31)
It t
Let us employ the following notation: Qi = ReQi, Qi = Im Qi. We
then have
P2 =
a^Q,
1-J«> +bl«
2 •
(32)
n > /!„
Since ng >> 1, we may change from the sum to the integral
Si k,v
^'~~Y^"^'
e "s^Tc
^--^Q\]+i^iQ\
(33)
We may disregard the quantity Q^ in the denominator of the integral,
and for Qi we may employ its value in the case of a '^ 1. In addition,
making an exponentially small error, we may multiply the lower integration
limit by <=°. We finally obtain
P2=.
l_e *3"t
(O3
e=i--^Q:/,-...
The dispersion equation (25) now assimies the following form
(l^zl£cihk.a +
. 2 1
^3f„
Yii
(1 - n
1
i\2
(34)
(35)
Let us assume that ks ->- ks = ks + 16 , 6 << ks. Then the real part /79
of equation (35) produces a relationship between the phase velocity g and
70
the frequency of the propagated wave w:
I
and the imaginary part produces damping:
S = -?= . ^3"t< _ (1— 32)(1— e) ^
Yti ' « 'p2e2(1 — e) , Mik^a'l^' (37)
Formula (37) can be considerably simplified in the case of large
(kj_ a >> 1) and small (kj_ a << 1) layer thickness. We have excluded
3^ from equation (36) , and shall substitute it in (37) . In the case of
k_|_ a >> 1, we obtain
V~^' <- '{\-\^\f'l^' (38)
where k = — . The damping determined by this formula coincides with the
damping found in (Ref. 5, 6) for a reflection coefficient of P = 1. In
the case of kj_ a << 1, we have
8 = -f. kv.e 1^^ • (^' + ^W? . (39)
It may thus be seen that wave damping is increased when there is a
decrease in the plasma layer thickness.
Ion-Sound Waves (v^^g >> V^ >> 'Vr^^)
The dispersion equation (25) may be considerably simplified, if we
set the speed of light c = 0°. In this case, we have
A = -^dl+i l-T,§iQi + b'Q2+2Q,)A. (40)
I L a={ -if
A A
We shall employ the following notation: A = Re Oj A' = I™ v2" /80
1C5 -kg
Then the real P3 and the imaginary Pi^ components in the sum (25) have
the following form
p - 2 y' A' .
n=0
4"* + A'" (41)
^* = i2x^- (^2)
n=0
71
We shall employ the same procedure in computing the sum P3 as for
computing Pi, but we shall take the fact into account that in the case
of ■; << 1 we have
kgVTe
We obtain
Re FQi {q) + i Q2 (9) + 2Q3 (<7) I = -2 -^ ■
t Je " "re
^3=^
(43)
(44)
where SS^ Q? m,
x^ = ^Hl+?); | = ^^:3,. = 1-J; , = ^.
Employing the integrals (23) , we find
Im (Q, + i^Q^ + 2Q3). = - -?— ^ exp f - ^ .
(1 + b^r^ ^
Since ^ , , 2 '^'^ ^ fo^ ^'^y '^j we then have
(45)
a — f . — ^
(1+6^)2
IfeF-^'n^fe)"}
Consequently, both electrons and ions make a contribution to the wave
damping. The ion component P^. of the sum Ptj has the same form as P2 with
a replacement of the indices e ->- i :
In computing the electron component Pi^ of the sum P,^, and in computing /81
P. ., we shall set A"^ » A'^. We then have
4 yT 1 — 5< 1
«-v^ + ^+.
(48)
Thus, the dispersion equation (25) has the following form for ion-
sound waves
^•^^ + /(P« + P.) = 1. (,9)
We may thus find the expression for the spatial damping decrement
72
&,
0-A:3(l+5) T;^— • (50)
m-
sh^xa
In a nonconflned plasma, where there Is no surface wave, the damping of a
three-dimensional ion-sound wave is determined by Cherenkov absorption of the
wave energy by plasma electrons, and the ion contribution to the damping
is exponentially small. As may be seen from formulas (47) - (50) , plasma
ions also make a great contribution to the damping for a surface ion-sound
wave. The physics here is the same as for high frequency oscillations —
the Cherenkov absorption of the wave energy by plasma ions is particularly
significant for the short wave components of Fourier expansion of the
surface wave.
Let us study equations (49) and (50) in special, different cases. If
1 /p
k3a(l + O » 1, then we obtain from equation (49)
b2-^.^^; (51)
3 y2 ,_2
"re I
4
/i ■ v^^.
8 =-h
vL |8,|ll-e,- )-| <" e,- 4p.«; ^ • J
kaa
2lT
(52)
since for large ksa the sum included in Pj^^ approximately equals -J—
It can be seen from formula (52) that the ions make a particularly /82
1/2
large contribution to the damping, if e. -^ -1. If k3a(l + ?) << 1,
then ^ , „ '
^[-(■-'fT]
(53)
^ = ks\^^{P,i + PA>). (54)
The following limiting cases are possible: (1) At a large plasma
electron temperature or a small layer thickness, when — Se,— ^ < 1 (| ^| <C 1),
the damping decrement is
8:
9
ta
e,- ^ jt<o
V-.
—2 ' 3
a' I e, l^j
(55)
(in the case of ksa « 1 the sum included in Pi^^ approximately equals y) '>
(2) For a large plasma layer thickness or an electron temperature which is
not too high when _ Se, rL_ ;^ l (^ ~ _ 1),
73
!t
-y ^ ^J .~[^-ef+/-8.Q3j (56)
We cannot assume that a -»■ <» here, due to the condition of the initial ex-
pansion.
As may be seen from formulas (52), (55) and (56), the damping decre-
ments of a surface ion-sound wave are large in different cases. Plasma
ions make a significant contribution, and sometimes the main contribution,
to the damping. The wave phase velocity is decreased with a decrease in
the layer thickness ; therefore , the ion contribution to the damping in-
creases, while the electron contribution decreases.
REFERENCES
1. Shuman, W. 0. Z. Phys . , 128, 629, 1950.
2. Fainberg, Ya. B. CERN Symposium, 1, 84, 1956.
3. Pyatigorskiy , L. M. Uchenyye zapiski Khar'kovskogo Universiteta, 49,
38, 1953.
4. Fajmberg, Ya. B. , Gorbatenko, M. F. Zhurnal Teoreticheskoy Fiziki, /83
29, 549, 1959.
5. Gorbatenko, M. F. , Kurilko, V. I. Zhurnal Teoreticheskoy Fiziki,
34, 6, 1964.
6. Romanov, Yu. A. Radloflzika, 7, 242, 1964.
7. Landau, L. D. Zhurnal Eksperimental'noy i Teoreticheskoy Fiziki,
16, 574, 1946.
8. Aleksin, V. F. In the Book: Plasma Physics and Problems of Controlled
Thermonuclear Synthesis (Fizika Plazmy i problemy upravlyayemogo
termoyadernogo sinteza) , 4. Izdatel'stvo USSR, Kiev, 1965.
74
KINETIC THEORY OF A SURFACE WAVE IN A PLASMA WAVE GUIDE
M. F. Gorbatenko, V. I. Kurilko
As was shown in (Ref. 1), when a surface wave is propagated along a
plane boundary of half-space occupied by a plasma having a temperature
which is different from zero, Cherenkov absorption of the wave energy by
thermal plasma electrons takes place. This leads to damping of this wave
even when there are no collisions. In contrast to the damping of a longi-
tudinal wave in an unconfined plasma, in this case for small thermal
velocities the damping coefficient is proportional to the thermal velocity
of plasma electrons.
We investigated this phenomenon for the case of a plasma wave guide
produced by a plasma layer having a finite thickness (21, |x| < 1).
Since the electrons moving at a thermal velocity are successively reflected
from both walls of the wave guide, it was not known previously that in
this case the absorption investigated in (Ref. 1) will not decrease con-
siderably.
The initial system of equations consists of the kinetic equation for
a high frequency addition to the distribution function and of a Maxwell
equation:
df
aT + ^vr/-^^vr/o = o
dt
rot^ = l!57 + l.p
c -^ ^ c dt
TOtE
c ' dt
J= — e J vfdv
(1)
It is assijmed that the equilibrium distribution function is a Max- /84
well distribution: f^ = —.
exp
vl + v^^
(the z axis passes along the
direction in which a surface wave is propagated) .
Let us write the solution for system (1) in the following form
f{x, z, t) =f (x) exp [i {kaZ — wt)], E {x, z, t) =
— E {x) exp [i (fegZ — 0)01-
We then obtain the following equation for the distribution functions of
particles moving in positive (+) and negative (~) directions along the
X axis
75
»± 9»f. ^ .- (2)
• /u)*<p± ± vj-l- + ?^ (y.e. ± «,£,) = D.
We shall employ the following expression as the boundary conditions for
the distribution function (|)* (x) in the planes x = + 1 of a plasma layer
c?+ (x = — /) = pf~{x = — /),
cp- (a: = /) = pf'^ (x = /)•
„+,._A (0<P<1) ^^^
The solution for equation (2) which satisfies the boundary conditions
has the following form
+ P ^ {v,E^ — v^E^) exp r^ {x' + X)] dx' — p^ f {v^E^ + v^E^) x
Xexpl-^{x'-x-2[)]dx'\; (4)
'P"^^^ = ;^A { I ^"^^^ - '^^^^ ^^P [^ (^' - ^ - 20] dx' -
- p j" (y,E. + t;;,£^) exp F- J^*(a;'+;c)1 dAT' - p^ [ (v^E,- v,E,) x
X exp l'^{x' —x + 2/)l djc' 1 ,
where /85
no — plasma electron density.
By substituting system (4) in the expression for the current, and
by substituting the latter in the Maxwell equation, we obtain a system of
two integro-differential equations for the E^ and E^^ fields. In the
general case, this system is cumbersome, and it is difficult to obtain an
analytical solution for it. However, in the case of slight thermal
scatter (v,j, -^ 0) , a solution may be found for the system. In actuality,
in this case, as may be readily shown, current corrections due to the
electron thermal motion decrease with an increase in the distance from
76
the surface of the plasma layer proportionally to
exp[-i?v.|i_^|'A]^^=^^, 7j=^].
Since there are significant thermal scatter phenomena only in a layer
having a thickness on the order of _Z. close to the plasma surface, they
may be taken into account by means of effective conditions which the
fields determined according to the Maxwell equation for "a cold" plasma
must satisfy.
The effective boundary conditions follow from the Maxwell equations,
in which the currents are expressed by the fields by means of system (4) .
For purposes of definition, let us investigate a wave in which the longi-
tudinal field component is symmetrical with respect to the plane
X = [E^,(,-x) = E^Cx), EL(-x) = -H , (x) ] (Ref. 2). As would be expected,
the currents J^ and J^ — according to system (4) — retain the symmetry
of the corresponding fields. Therefore, it is sufficient to obtain the
boundary conditions for one of the plasma layer surfaces; they will be
satisfied on the second surface due to the field symmetry. Let us find
the boundary conditions which must be satisfied by the tangential field
components on the x = 1 surface. Integrating equations containing J^ and
J^, over an infinitely thin layer e(Z2. << e << 6, 1, where e — layer /86
thickenss, 6 — depth of skin-layer) , we obtain
i—t
When the integrals in equations (5) are computed, it is advantageous first
to perform integration over n :
Xexpf-f (l-e-V)ldV-
I— I L J l i_£
L J 1— •
X exp
77
X exp {— $2
"o^ '
){j£.(V)expr-'|(l+V)ld7,'-
- Jf.hOexpFf (l+V)ldV|;
; = .jfexp[-^^-fjJ^.(,Oexp^-fx
V)j rfV - /P^ J f exp 1^ - S^ + 2^1 J £, (Y) X
K exp P| (1 - e - vj') jdT]' + tp j' f exp (- ^2) { j £, (71 ') X
L Jo _i
I
xexpf-f (1+V)W- |£^h')exp pi {l+V)]rfV}.
X (1-e-
(6) /87
where only terms making the largest contribution in the case of Vr^ ->
are retained. Performing integration over n' in equations (6), we find
2iR 2iR
asr,
R
2iR 2
7«- +
e ^ — p^e ^
r _£^ _lR-ri
■ +|^£^(l)exp(-^^)^^d$32« 2^;
e , ^ — p^e ^
2i;? 2i"R
i n\ »
»0
2 /: = ^'Iexp(-$^)W^1Ip i£«
2
_ J J £^ (1) exp (- ^2) \d% -37W i«
e ^ — p^e ^
(7)
Since J^^ and J^ in equations (7) are proportional to Vrj,, when the integrals
are computed with respect to C = — ^ in these equations, it is sufficient
to confine oneself to the zero-th approximation with respect to v (i.e., to
78
strive to zero in the vj intervals) . Taking the fact into account that
Imo) > 0, we finally obtain
u>l(l~p)Ei{\);
N»i
(8)
Thus, the effective boundary conditions taking into account the
thermal motion of plasma electrons have the following form
2 ]/^o>2/?
««o(l+P)
, W-. « .. KlUif. II -]- I/I IO)« II -1- Wl _,, I
(9)
Substituting the solution of the Maxwell equations for a plasma
( I n I < 1) in a vacuum ( | n | > 1) in these boundary conditions
ik^a
where
E'^ (71) --=Ach-Uri; E'J {-q) = -J A shxlri; H^ (vj) = -'-^A sh^l-q,
2
t: — «3 — e3« , S3 = 1 — -2 , « = — ,
(10)
/88
we obtain the dispersion equation
S3 [X + .;. th X/] = ^; [(I + P) ^1 th X/ - i|l (, _ p J
(11)
In the case of 1 ->■ °°, this equation changes into an equation for half-space
(Ref. 1).
Thus, in the case of a plasma layer having a finite thickness, the
surface wave damping is proportional to v.^. However, the wave damping in-
creases with a decrease in the layer thickness , because the wave phase
velocity decreases :
AA3 =
KiK„s|/3„3
(1+P)kl,l'+^-^
; hal»l, e3<0;
(12)
79
REFERENCES
1. Gorbatenko, M. F. , Kurilko, V. I. Zhumal Teareticheskoy Fiziki,
34, 6, 1964.
2. Vajnishte3m, L. A. Electromagnetic Waves. "Soviet Radio" (Elektro-
magnitnyye volny. "Sovetskoye radio"). Moscow, 1955.
SINGULARITIES OF AN ELECTROMAGNETIC FIELD IN A /89
NONUNIFORM, MAGNETOACTIVE PLASMA
V. V. Dolgopolov
As is well known, in an isotropic, nonuniform transmittant medium there
is a sharp increase in the electric field strength of an electromagnetic
wave at the point where the dielectric constant vanishes. This increase is
caused by plasma resonance. This phenomenon, which has been called "infla-
tion" of the field has been studied in several articles. It is pointed
out in (Ref . 1) that the "inflation" must occur in a magnetoactive medium
at the point where the refractive index for the wave becomes infinite. The
phenomenon of field "inflation" for normal wave incidence on the layer was
studied in (Ref. 2) for the case when a constant magnetic field was parallel
to the plasma boundary.
This article investigates the behavior of the electric field of an
"inflated" wave close to the "inflation" point in a magnetoactive plasma
for the case of an arbitrary angle between the direction of the constant
magnetic field and the normal to the plasma layer surface, and for the case
of oblique wave incidence on the layer. A solution is found over a wider
region for certain special cases for a wavelength which is a little less
than the distance at which the layer parameters change significantly.
We shall assume that the parameters of the plasma magnetoactive layer
change along the x axis, and the vector of the constant magnetic field lies
in the xOz plane, forming the angle S with the z axis. The tensor of the
dielectric constant e^j^ can then be written in the following form
''ejcos^ + sgsin^ 9 kzCosQ (63 — ei)cos9sin ©N
— /eg cos 9 , Ej /e2 sin ' ^^
(£3 — e^) cos sin — /sg sin Sj sin^0 + S3 cos^
80
e/ft =
The quantities e^, e2» £3 will be real, if dissipative processes are not
taken into account.
Assuming that the dependence of the electromagnetic field strength
on time, y and z, is determined by the factor ei(-(jdt + k^y + k^^^ » ^^
obtain the following system of equations from the Maxwell equation for
the electric field components
(k^ - i ^11) E, + i (ky ^-^ + k^^-^) - ^ (e,,Ey + e.gE,) =
— (kyk^ + '^^2^E^ = 0\, (2)
iku
/yfe^--e E
dx
{kykz + ^ HijEy +
where k^ = k^ + k^.
y z
/90
The following inequality is fulfilled in the vicinity of the "inflation"
point for an "inflated" field
dE
dx
>k
This enables us to represent the field E in this region in the form of a
series
E = £(0) + £(') +
where
EO)\-^\kx\\E('>^^\E<'>^\
We shall always select the origin along the x axis, so that x = at the
inflation point. We obtain the following from the two lower equations
of system (2)
£(0) _ £(0) _ 0. -^ = ikyEW + Cy.^=^ ik.EW + C;
- I = ii^ f_
dx'' ^ dx c
dx ~ ""^"^^ ' ""' dx
where C and C^ are the integration constants,
81
Substituting the expressions obtained in the first equation (which
is differentiated with respect to x) of system (2) (we shall disregard
components containing ^ and ^^ ) and taking into account (1) , we /91
8x 9x
can write the equation for e( ):
X
I (HxE^^) + i2k,H3Eio, ^ const = 0.
It can be seen from this equation that "inflation" can only occur where
e^l vanishes. In the given approximation, it must be assumed that £^3
is constant, and is limited to the linear component in Taylor expansion
for Ell. Then the equation for eC^) assvraies the following form
where
11 dx \ x=o '
We can write the solution of equation (4) in the form of a sum of
particular solution and the general solution of the corresponding homo-
geneous equation. The particular solution is a constant quantity, and Is of
no interest. The general solution of the homogeneous equation for the
field E close to the "inflation" point yields the following expression
c- ^ „— :i In (kx) (5)
where A is a quantity which is constant with respect to x;
0=2^.
If a f= 0, we have
^y
where C^ and C2 are integration constants.
In the case of a = 0, we have
Ey = ikyA In {kx); Ez = ik^A In (kx).
82
Thus, if kg. = or e^ 3 = 0, the nature of the field "inflation" E
is the same as in an isotropic medium.
If k2 T^ and e-^^ i= 0, the closer the "inflation" point, the more
intensely does the field E oscillate on both sides of it. Thus, the
amplitude of the field component E increases as — , and the amplitudes /92
of the components E and E remain finite. It can be seen from equation
(3) that if ^^ = 0, at the point where en = "inflation" does not
3 X
occur in the case of k e , f= 0.
Let us investigate the case when the medium parameters change slowly
and the method of geometric optics may be employed when solving the system
of equations (2) outside of the vicinity of the "inflation" point
X
E =A (x) e ° ' ^
Substituting expression (6) in the system (2) , we obtain the following
equation of the fourth power for x(x)
eiix" + 2fe,si3x3 + a W x^ + P (X) X + Y W = 0. (7)
We shall not give the expressions for the coefficients a, 3, y, due to
their ciombersome nature.
If k^. and e-^g are different from zero, we have the following from
equation (7) for the wave vector component x(x) of an "inflated" wave
close to the "Inflation" point
y~ - 2k^ "^' .
This result coincides with formula (5) . The wave oscillates in space on
both sides of the "inflation" point.
1. If the magnetic field is parallel to the plasma layer surface
(5=0), e,_ and B(x) vanish and equation (7) becomes a biquadratic equa-
tion. We than have
^2 _ k2 ii+i3 A.5 _ f! . /' -.^^l!i±!?^ +
(8)
±i;^[i^^-^^)[K-ii^ + i^l]' + ^'iK4^^-
83
Taking into account the components of subsequent order in the expansion
with respect to the small parameter-r — /x2 in the system of equations (2),
dx ^
we obtain the expression for the vector components A(x) :
* z r2. ■
K] '
(9)
where C is the integration constant corresponding to one of the four /93
functions x (x) ;
X^ky (S+C+ + r C-) + k. (C+ + C-) -/) - 2x ($+$- + ^^);
^ = ^' [>^l - ? ^i) -c^ ^r- K^ = x^ + A^ - ^e,;
If we may disregard dissipation, in the region where x is real, Y and Z
are real, and the exponential factors in equations (9) characterize only
the wave phase.
As may be seen from (8), when e^ strives to zero (in the case of
S=Oeii=ei) one value of x^
comes infinite as J^:
goes to a finite limit and the other be-
1
£, =
■k^ + H-2
7^-2
4 + K
'i^[^i^^+U\
t,—
In the case of x2 = ^-^i, the solutions of (9) are valid for the point
where ej = 0. They are not "inflated". The remaining two solutions of
system (2) for the vicinity of the point where ex = cannot be represented
in the form of formula (9). When trying to determine them^ one may assume
that £2 and E3 are constant, and for ei one may restrict oneself to the
linear components in the Taylor expansion:
t^x.
After several ctmibersome computations, we obtained the following
equation from the system of equations (2) for the field component E
84
Disregarding the last two components, we may write
d?u 1^ 4 du 1 „
W^T-^-Tx — Y^^"' (10)
where /94
«=-a^: p = ~— -. .
If we substitute u(x) = C~^v(5), x = ?^, equation (10) becomes a Bessel
equation
and, employing the recurrent formulas for the Bessel function, we obtain
where A and B are the integration constants; Ii (2/ px) and K^ (2v'px) —
the McDonald and Bessel functions. In the region under consideration,
the components of the field E are related by the following relationships
which enable us to determine E and E^, :
E^ E
^y=k;-~\Au{2Vrx)-BKo{2VVx)]. ^^2^
Taking the asymptotic form of the Bessel function and requiring that ex-
pressions (11) and (12) change into expression (6) for large values of |x| ,
we may find the relationship between the coefficients A and B and the co-
efficients C included in (9). In the case of p > 0, the coefficients C
are related to A and B in the following way to the right of the "inflation"
point (x is imaginary in this case):
85
-+•
{-
1 /"'^iV'i^— JtS
where C = C^ in the case of x imaginary negative; C = C_ in the case of
X imaginary positive.
2. If the magnetic field is perpendicular to the layer surface /95
IT
(S = — ) , the system of equations (2) may be reduced to a system of two
second order equations
■i
(13)
where
^. = p (ky^ + k^f): £z = i {k^ - kyf); E. = -^-i . ^
-A'
dx'
Outside of the vicinity of the "inflation" point, and representing (j)
and ijj similarly to (6) in the following form, we have:
X
<5 -ndx
?,'!' = A,, ^{x)e ° ,
We obtain the following from the system of equations (13) in the approxi-
mation of geometric optics
where
Q =
0)2
r2 ^3 ■
■k'
[(^^+^4)^3-^ft'=4]
K'
= .^+A^-^e,;
86
C is an integration constant corresponding to one of the four
ftmctions x(x).
As may be seen from (14), close to the point £3=0 (in the case of
T .
the following form
= -rEii = £3) one solution for x'^ is finite, and the other solution has
In the first case, solutions of (14) are valid at the point £3=0.
They correspond to a "non- inflated" wave. In the solution of system (13), /96
we shall assume that ei and £2 are constant, and we shall expand £3 in
Taylor series in the vicinity of the point £3=0, and shall confine our-
selves to the linear component
We may find the following from the first equation of system (13)
(15)
Substituting this expression in the second equation of system (13) ,
disregarding the small components, and making the substitution v =
-7~T > we obtain
dx2
d^v , \ dv . V „
dx^ '^ a: dx ^ "^ X
where
3
When 5 = 2/ px is used as the Independent variable, this equation
is reduced to a Bessel equation
a-o , I av , f.
Utilizing (15) , we find the following from the latter equation
if = Ci//^" (21/^) + CM^ {2VTx), (16)
where Cj, C2 are the integration constants; H^J^ (g) , h(§) (?) — Hankel
functions .
87
As may be seen from (15), close to the "inflation" point (fi | 'v- 1 e 3ij; | <<
<< 1^1 — i.e., the contribution from (|) to the expression for the fields
in this region is negligibly small. If we make the stipulation that (16)
change into an expression like (14) (outside of the vicinity of the "in-
flation" point) J we may find the relationship between the constants C2 , C2
in (16) and the constants C in formula (14) .
Just as in the case of an isotropic layer, the presence of "inflation"
points in a magnetoactive , nonuniform, transmlttant medium leads to the
fact that the thermal radiation intensity of this medium is on the order /97
of the radiation intensity of an absolute black body, if the wavelength
being studied is comparable to the distance from the "inflation" point
to the layer boundary. When an electromagnetic wave, whose length is
comparable to the plasma layer thickness, falls on this layer, the amount
of energy absorbed by the plasma in the vicinity of the "inflation" point
per unit of time (per 1 cm of layer surface) is
^p2 (17)
where E is the electric field amplitude in a vacuum.
4jt
However, (17) is valid until the phenomena caused by the oscillation
nonlinearity and spatial dispersion in the vicinity of the "inflation"
point exceed the phenomena caused by particle collisions in this
region, i.e..
l(^V)y|:^|vy|; kY
where v is the electron velocity caused by the wave field; v — the effec-
tive frequency of electron collisions with ions; K '\- — '— — effective
wave vector in the vicinity of the "inflation point; SL — layer thickness;
m — electron mass; T - temperature. This leads to a limitation for the
field E and the temperature T, which can be fulfilled by (17) :
where Hg — the constant magnetic field strength; L — Coulomb logarithm;
nQ — electron density in the plasma; e — electron charge. The maximum
energy obtained by a charged particle per unit of time is
W TTT-J •
lOm'c/Io
88
In the case of hq = lO^^ cm-3, T = 10^ "K, Hq = lO^G, co 'v 5-10
sec"-"^, w 'V' 100 ev/sec, E ;^ 1 v/cm.
10
Thus, it is not advantageous to employ "inflated" fields for plasma
heating .
REFERENCES /98
1. Ginzburg, V. L. Propagation of Electromagnetic Waves in a Plasma
(Rasprostraneniye elektromagnitnykh voln v plazme) . Fizmatgiz,
Moscow, 1960.
2. Denisov, N. G. Radiotekhnika i Elektronika, 1, 732, 1956.
EXCITATION OF A MAGNETOHYDRODYNAMIC WAVE
GUIDE IN A COAXIAL LINE
S. S. Kalmykova, V. I. Kurilko
As is known (Ref. 1), magnetohydrodynamic waves have low frequencies
(to << fixT ^ 1.5'10^ , with respect to A,,^^ >> 10^ cm). At these fre-
n v/ sec vac
quencies, a coaxial line represents the most reasonable method for trans-
mitting energy from an oscillator to a wave guide. Therefore, the problem
of excitation of an axially symmetrical E-wave in a magnetohydrodjmamic
wave guide by a TEM wave of a coaxial line is of great interest. If the
plasma temperature may be disregarded, and its conductivity is great enough,
under these conditions, the plasma may be characterized by the dielectric
constant tensor without spatial dispersion. The problem under considera-
tion is then a special case of the problem regarding the matching between
an anisotropic dielectric wave guide and a coaxial line. Equations for
determining the stray field were obtained in (Ref. 2, 3):
'? ^'^ + 2 (nif j- [z in + z„ (/')] (f - J L^o ('") 2o C) J '' - 1'
(1)
~w^ j"
+-" z,{nz(t' )dt' . inj^x)
[z (/') + z„ (/')) {i'^ - k') {(' - • ^
89
The diffracted magnetic field in the space between the wave guide
(radius a) and the casing (radius b) is expressed by means of the in-
determinate function (j>(t) E (})+(t) - (|)~(t) = <|)+(t) - <i>'^(-t) (the indices /99
(+) designate the analyticity in the upper or the lower half plane of
the complex variable t) :
+~
//(z)== Y- • ^r-! • —I ?— r-(''+(0-4-9+(0-5^ X
X ^"zr;^) exp (itz) dt for- 2 > 0,
(3)
where
x+(/)_x+(-/) = -^<p(0 +
7 if\^^ ^0 (a) . 7 //■> _ P (0 MM .
A„(r)=/„(yr)^o(y6)-(-l)''/Cx(yr)/o(t;6), «=0,I;
= (/2 _ /fe2)'A; p =
";J-(^^-^^sJJ^
If the plasma density is large (uga^ » c^), equation (1) can be
solved according to the iteration method.
The reflection coefficient of a coaxial wave in the first approxima-
tion, as a function of the parameter — - — - ■■ , , is
a In(b/a)
(4)
When the amplitude of three-dimensional waves excited in a magnetohydrodynamic
wave guide is computed, a distinction must be drawn between the four regions
of the plasma parameters and the wave parameters . The magnetic field
amplitudes have the following values corresponding to these regions (the
amplitude of the coaxial wave equals unity) :
'^'^=Eh)^^n + ^^-Hbla)\-\ 3,-l«-^«l;
90
II I I I II I
-T'«e^-l«l:
2 r >.y J
^"=5n!^N- -?-«•«..; (5)
Thus, the effective excitation of magnetohydrodynamlc waves by a
coaxial line is only observed for strong magnetic fields (ej_ - 1 << 1) .
2
Harmonics with large numbers corresponding to the limiting case X >>
>> a^/6^ are always only slightly excited. For weak magnetic fields, the
main portion of the coaxial wave power is dispersed into excitation of
the surface plasma wave.
REFERENCES
1. Glnzburg, V. A. Propagation of Electromagnetic Waves in a Plasma
(Rasprostraneniye elektromagnitnykh voln v plazme) . Fizmatgiz,
Moscow, 1960.
2. Kalmykova, S. S. Uspekhi Fizicheskikh Nauk, 9, 2, 1964.
3. Kalmykova, S. S., Kurilko, V. I. In the Book: Plasma Physics and
Problems of Controlled Thermonuclear Synthesis, 4 (Fizika plazmy i
problemy upravlyayemogo termoyadernogo sinteza, 4). Izdatel'stvo
AN USSR, Kiev, 68, 1964.
/lOO
THEORY OF MAGNETOHYDRODYNAMIC WAVE SCATTERING
AT THE END OF A WAVE GUIDE
V. I. Kurilko
The study of magnetohydrodynamlc waves is of great interest in
solving several problems of plasma physics, such as controlled thermo-
nuclear synthesis, magnetohydrodynamlc oscillators, etc. (Ref. 1). A
great many articles have been recently published which investigated the
91
Illlllilll
propagation of magnetohydrodynamic waves in unifoirm, unconfined magneto-
hydrodynamic wave guides. However, practically every wave guide is con-
fined. Therefore, it becomes necessary to investigate the phenomena re- /lOl
lated to the scattering of magnetohydrodynamic waves at the end of a
wave guide (for example, reflection of one of the eigen waves of such a
wave guide, and its transformation into other waves). In addition, the
theory of electromagnetic wave scattering includes the excitation of a
confined wave guide.
This article investigates these phenomena for a semi-infinite
plasma wave guide. In the general case, this problem may be reduced to
a system of two coupled integral equations or (in the presence of an
infinite casing) to two infinite, coupled systems of algebraic equations,
whose solution may only be found by numerical methods. Therefore, let
us investigate the case when the end of the wave guide is covered by a
conducting diaphragm. As will be shown below, the problem of determining
the Fourier component of *]\e scattered field may be reduced to an integral
Fredholm equation of the second type, whose solution may be obtained by
a numerical method, and when there is a small parameter — in analytical
form.
We shall assume that the plasma conductivity is infinite, and its
temperature equals zero. In this case, the electrodynamic properties
of the plasma wave guide, as is known, may be characterized by the di-
•, . "n ■, "0/2 Aire^n eHn
electric constants ej_= 1+ ^, e „ = 1 - _ (^a,J = —^, '^ = ^,
JL, = >> 0) I , which only depend on frequency. Since the frequency is
^ Mc /
fixed in our problem, the plasma wave guide may be regarded as a special
case of an anisotropic dielectric wave guide. Therefore, let us first
investigate the more general problem of electromagnetic wave scattering
by the jump in the dielectric constants of an anisotropic dielectric wave
guide, whose uniform sections are separated by the conducting diaphragm.
The plasma wave guide parameters may be employed to find the analytical
solutions of the general equation obtained.
Thus , let us investigate an anisotropic dielectric wave guide
(r < a, -<=° < z < +») with a piecewise-uniform tensor dielectric constant
e(z>0. 0<r<a) = {eO), eO));
s(z<0. 0<r<a) = (e(2). e<?));
A
e(— oo<z< + co, a<r <b)^\.
92
Let us assume that the uniform sections of the wave guide are /102
separated by an ideally conducting diaphragm (z = 0, < r < a), and
that the conducting casing is not confined and is located at the distance
r = b from the wave guide axis. Let us assimie that one of the eigen,
axially symmetrical E-waves of this wave guide, which is characterized
by the wave number hjjj, falls on the nonuniform section from the right
wave guide. Let us determine the amplitudes for the eigen waves of
both wave guides which are excited due to scattering of this wave. We
may write the solution for the fields in the following form
+ "
//« = ^ //(OAi {t)e"'dt + exp{~ih„z)
\
£| = j" f //x (0 A„ (0 e"^ dt + Zl exp {-ihrnz)
— oo <2<
< + oo;
t^n(t) = ln(va)KAvb)-{-l)''Kn{va)U{j:b), n = 0.1:
r = a + 0,
(1)
u (0 = (/2 — A2)v.; k = '^, lmo>>0;
Hl = \K (0 h (PsQ) e^'dt + 2A, cos A„z
E\ = t > (0 hs (0 Ji (Psa) e^^dt + 2A,Zl cos hmZ
?; =. J ^ /j. (0 Ji (Psa) e"' + 2Ash„ (sin h^z) ^
CO ^ )
r = a — 0, ,
s = 1 — z> 0,
s = 2 — 2 < 0,
(2)
where
Ai = 1; Ai =0.
The fields thus selected satisfy the Maxwell equation. We can deter-
mine the remaining, unknown Fourier amplitudes H, hg of the desired fields
from the boundary conditions on the lateral surfaces of the wave guide /103
and on the conducting diaphragm. The boundary conditions on the wave
guide — vacuum surface have the following form
93
lilllli
//y' = '^%' ^i'* = £^ f or r = a, 2 > 0;
Hf = //J. £f = £', for. r = c. 2 < 0. (3)
Substituting (1) and (2) in the boundary conditions (3) and employing
the results derived in (Ref . 2) , we may express the vinknown f vinctions
H and hg by the boundary values on the contour Im t = of the functions
which are analytical in the upper (+) and the lower (-) half-planes of
the complex variable t :
K it) A (ha) = ^) [z it) T+ it) - 1+ (0 - 2^.7(7-^!)} '
h. it) A (M = ^ {z (/) X- it) - r it) - 4r^} ' ("^^
// (/) Ai it) = ^1^^ [z^ it) <p+ (0 - ^+ it) - 2.V(!Im 1 =
D,it)~Z^it)-Zit), Z(0=f • J^.
where
The latter equation (4) represents a boundary problem for determining
the unknown functions (j)"*", \^'^, K~ and x~. The relationships lacking between
these functions may be determined from the boundary conditions on the con-
ducting diaphragm
£^(2 = 0; 0<r<a) = 0. (5)
Substituting E^ in the boundary conditions (5), we obtain
h'i—t)^h'it). (6)
We obtain the following by means of the latter equation and the Sommerfeld
condition for the finiteness of the magnetic field and integrability of
the electric field close to the diaphragm edge:
^-it)~-<ffi-t); S7(0=-t^(-0. (7)
where
/104
94
Im hm > 0.
(Equations (6) are only necessary for fulfilling the boundary conditions
(5). We shall assume that these equations are sufficient, although this
may only be proven for a rectangular wedge [Ref. 3]).
Thus , the boundary problem for determining the function assumes the
following form
(8)
This type of boundary problem for one special case (ei = 1, £2 ^ ")
was first studied in (Ref. 4). It was shown that in the presence of a
casing, when the coefficients have singularities of only the pole type,
it can be reduced to an infinite system of algebraic equations. However,
by employing the formulas of Sokhotskiy — Plemel' , it is more advantageous
to reduce the problem (8) to an integral equation. By combining and sub-
tracting equation (8) with its mirror image at the point t = 0, we obtain
(Ref. 5)
1 (hsn_^ , _L z / A V ^(\ JL_ _ 1 ]h (n ^.. _
2 (9)
A„ Z (t) \y Z' (t) — Z'
--if -,-13^2, o[^t) ^ /W= 2(0ti(0 = '1'x(0.
5=1
The index of the latter equation equals zero (Ref. 5). Therefore, it is /105
is equivalent to one integral Fredholm equation of the second type for
'J^l(t) = .j^l(t) - i^+(-t):
95
+ - 2 + =
(10)
S=I
ij^,^/r)
According to the general theory of singular equations with a kernel
of the Cauchy type (Ref. 5), the solution of (9) which vanishes in the
case of t ->• " exists and is real. Due to the equivalence between (9) and
(10) , the same holds for the solution of equation (10) . It can be solved
numerically in the general case, and in the presence of parameters — it
can be solved analytically, even if there is no casing (b ^ «>) and the
coefficients in equation (8) have singularities of the point branching
type.
In the case of the magnetohydrodynamic wave guide, which we are in-
vestigating, the ratio between the wave guide radius and the wave length
in a vacuum serves as one of the large parameters. Even for Hg '^ 10^ gauss,
Xq ^ 7^ '^ 10^ cm. Therefore, in the case of a ^ 5 cm the ratio a/X (A —
wavelength in a vacuvmi) cannot exceed 10"^ (with allowance for the require-
ment that X >> Xq), so that even In (X/a) is large (In (X/a) ^4). In
addition, in a significant number of important cases the linear plasma
density na^ is great, so that the ratio — = is small.
Assxmiing that the inequality 6 << a << X is fulfilled, we can signifi-
cantly simplify equation (10) . Disregarding terms on the order of /106
(6/a)In(X/a) , (— ) In(X/a) and taking the fact into account that £2 - 1»
•(f)
my t'-t '. Ki yzimt'-t)- r:i ' t'^_hi[+ z(t)\
we obtain , ,
(10')
(the index «: 1 » for i|) is omitted from this point on). Equation (10'), as
may be readily seen, is equivalent to the scalar boundary problem
<j.+ (0=z(/){"-(/)-^-^[i + ^]}. ^^^^
The solution of (11) has the following form
96
where
^■'(0 = f7S^-(0; ^+(0 = 3p^.
and the singularity at the point t' = t passes around the contour C from
below..
Within an accuracy of a term on the order of { — \ In — << 1, we
have \x; a '
(?)
X+ (t) = (to + kayf' exp fc C y=^. ' FT7V X
I ka^ (12')
xln[ln(|-nn|)]|.
where y is the Euler constant, and in the case of In — >> 1 we have
a
^"(') = [|-nn|]"'«» + .«)v.{.-|.!i«±f!).
1 xl"i
Expansion in powers of In — is usually employed in antenna
I ^ I
theory (Ref. 6), and numerical integration may be employed to determine
expression (12') for YX+(t) which are not too small.
In the same approximation (6 << a << A) , the boundary problem for de-
termining the function <j)+(t) has the following form
T+(0=,4^--(0-S'-^[l+^]. (13)
The integral in the solution of (12) may be readily computed by closing
the integration contour in the lower halfplane in the first term, and
in the upper halfplane in the second term, by the complex variable /107
t. The integral in the solution of (13) may be computed similarly for
(()+(t) . Substituting the expressions thus obtained for i('''"(t) and <|i"'"(t) into
97
J
i
the formula for the amplitude of the Fourier-field in a vacuum
Ax (0 ^(0 = 5;^ {2^ (0 ^+ (0 -1'+ (0 + ^ • ^-7^]' (1^)
by means of (1) we may find the expression for the amplitude for a wave
with the number 1, which is excited in a wave guide (z > 0) during the
scattering of a wave with the number m:
R,--^^'fi'-^^+^-
^ +
(15)
^/2;L 1 „ _km
ni,m^=^-
whereDu = ^D,|,= ft,;Z;„ = Z^(A,J;Xt„ = X+(A,„);
The latter expression may be significantly simplified in the most in-
teresting cases of small and large retardations njj^ ^. It thus appears that
the sum in the parentheses. in (15) is represented in the form of a power
series of the logarithms for the ratios X/a and X/6. Retaining the old
terms in this series, we obtain
^>Vr^''"^- (16)
^1 (v/. m) = 0; ni = Sj.;
r »S ]' In (e, - 1) ^
°('x-')
98
^«nx-l«l; (17) /lOS.
«*vf "j." a an'/'
In - In*
a an^
-^«l«nx«^;
Xa . 1 . X . X
(18)
In -jj In - -I- In ^- In
52 <$.>/, m <5. p- (19)
It can be seen from these formulas that the coefficients of the inter-trans-
formation of the magnetohydrodynamic waves Ri decrease with an increase
in the wave number and a decrease in the magnetic field strength. If the
plasma wave guide is surrounded by a conducting casing r = b, -«> < z < -H",
it is impossible to study magnetohydrodynamic waves in the case of X >> b,
and only the transformation of one oscillation into another occurs at the
end of the wave guide. The transformation coefficients for this case may
determined by substituting X+ (t) from the solution of the corresponding
problem X + (t) = valn^ X - (t) and the wave numbers h^^ui — from the
solution of the dispersion equation, in expression(15) .
D, {h,y. = Z\- ^' , > In I = 0..
REFERENCES
1. Artsimovich, L. A. Controlled Thermonuclear Reactions (Upravlyayemyye
termoyadernyye reaktsii) . Fizmatgiz, Moscow, 1963.
2. Rapoport, I. M. Doklady Akademii Nauk SSSR, 59, 1403, 1948.
3. Kalmykova, S. S., Kurilko, V. I. Doklady Akademii Nauk SSSR, 154, 6, /109
1964.
4. Jones, D. S. Proc. Roy. Soc, A215, 153, 1953; Philos. Trans. Roy.
Soc, A247, 499, 1955.
99
5. Noble, B. The Wiener-Hopf Method. Izdatel'stvo Inostrannoy Litera-
tury (IL), Moscow, 1962.
6. Muskhelishvili , N. I. Singular Integral Equations (Singulyarnyye
integral ' nj^e uravneniya) . Fizmatgiz, Moscow, 1962.
7. Vaynshteyn, L. A. Zhumal Teoreticheskoy Fizikl, 29, 673, 689, 1959.
DETERMINATION OF PLASMA TEMPERATURE AND DENSITY
DISTRIBUTION BY REFRACTION AND DAMPING OF A BEAM
V. L. Sizonenko, K. N. Stepanov
Let us Investigate the passage of a beam through a nonuniform plasma.
We shall show that, by changing the angle of incidence or the frequency of
the microwave signal, we can determine the plasma density distribution
by the beam refraction, and can determine the plasma electron temperature
distribution by the damping of the wave energy along the beam.
In order to determine the beam trajectory in a plasma, let us employ
the Fermi principle
8pds=0, (1)
a
where k is the wave vector; ds — an element of length along the beam tra-
jectory. In an isotropic plasma, we have
k^^y^^-wl, (2)
where o) is the wave frequency; ojo(i^) * W ^"^ — plasma Langmuir fre-
f m
quency; n(r) — plasma electron density. Modulation is performed in
formula (1) for fixed ends (^x^ a '^ ^^i b ~^' Taking the fact into account
that ds = vg^^dx^dxi^, from expression (1) , with allowance for (2) , we may
obtain the equation which determines the beam traj ectory for given wq (r) /IIQ
100
ds
I 1/~1 2 ^^A^ 1 1/^ 2 ^Skl <^>'k ^^i 5 -1/-^- 2 „
By employing equation (3), let us investigate the inverse problem:
Let us find wq (r) , i.e., the plasma density distribution based on the beam
refraction, as a function of the angle of incidence for a fixed frequency,
or as a function of the frequency for a fixed angle of incidence.
Planar Problem
If the density of a plasma filling the halfspace x > depends only
on one coordinate x, and it increases monotonically with an increase in x,
then the beam trajectory in the case under consideration is flat (Figure 1)
Equation (3) may be written as follows
^ ' (4)
•= Ydx'' + dz''
Integrating equation (4) , we obtain
]/ (1)2 — cBo^ = a = const, ^^^
where a = to sin )|;, ijj is the angle of incidence.
The position of the point at which the beam leaves the plasma z = I
can be determined from the plasma density distribution; it depends on the
angle of incidence and frequency. If the dependence i,{^) is known from
experiments, we may obtain the density distribution n(x) in the case of
X < X*, where aiQ (x*) = w (this problem is similar to the problem of deter-
mining the potential energy by the specific dependence of the oscillation
period on energy [Ref. 1]). In actuality, it follows from equation (5)
that
dz sirn]^
dx
/
<^os'*-^ (6)
We thus have
/111
(7)
101
where x = xq is the rotation point of the beam, determined from the
following condition
cos'' tb = i— .
* (1)*
2
Introducing a new variable u = ~ instead of x, from formula (7) we
obtain cos"<i>
It thus follows that
I 7 cos'iji
(8)
-^4i^L = +2\dcos«a ^^'"
sin-f-y^— cos2<j; J ^J K (cos2 4. — u) (f — cos'ii;/)
7
I
2^j|rf« = + 2u^(T).
2
too
Assuming that y = ~r» we finally find that
0)^
-^ ^^-=. (9)
COS^l}'
Equation (9) determines the function x(a)o) » i.e., the dependence of density
on the coordinate in implicit form, according to the specific dependence
l{ii) . We may employ this equation in the case of a cylindrical plasma,
when the plasma is probed by a beam in the plane ^ = const passing through /112
the cylinder axis. Thus, oig = wgCr) and x = r = R, where R is the plasma
radius (uqCR) =0). By performing similar measurements for different
values of the aximuthal angle ()), we may obtain the density distribution
in the case when density depends on the angle ^ (but does not depend on z) .
The plasma density distribution may also be determined by probing the
plasma with microwave signals having a different frequency. For a given
angle of incidence, the position of the exit point of the beam z = £ de-
pends on the frequency co. Knowing the function £((0) from experiment, we
102
Figure 1
may readily determine the function oIq (x) according to equation (7) . It
follows from equation (7) that
2<s>dx
\ COS^lJ''
= 'r:x{^).
(10)
0)0
Assuming that y= 7>
" COSlI)
(joq (x) in implicit form
we obtain the expression determining the dependence
costj/
X =
1 p / (a.) dai
(11)
We may also obtain formula (11) from the expression given in (Ref . 3)
[see also (Ref. 4)] for the real layer height x((0o) according to the specific
effective height Xg(w) for the case of normal wave incidence on the layer,
with allowance for the following relationship (Ref. 4)
l{'^)=2tg'^Xg{i>^cosj>).
Cylindrical Problem
If the beam passes through a plasma cylinder in a plane which is per-
pendicular to the cylinder axis (Figure 2) , equation (3) may be represented
in the following form
i(/-=']A;;^:r75|) = o; ds=.Vd?^T^^'
(12)
103
Figure 2
The plasma density distribution n(r) may be found from the specific
experimental dependence of the angle at which the beam leaves the plasma
(jjQ on the angle of incidence i|/ * (this problem is similar to the problem
of determining the potential energy from the specific dependence /113
of the scattering cross section on the scattering angle (Ref. 1, 2). Inte-
grating equation (12) , we obtain
sin^ <j» ,
(13)
It thus follows that
cpo(<}')=2/?sin^
rfr
(14)
where r = rg is the rotation point of a beam which is determined from the
following condition
Assuming that
* This problem was solved in (Ref. 5),
104
1 ^ (t "oU*
we can write equation (14) In the following form
stn'<|<
,p.(« = 2slntj j7j=g;y|
J y^u — sin* ^
(15)
from which we find that
1
1 1 sin»<(i
du
du
/(sin* i|/ — f) {u — sin* <}/)
Assuming that « = y* = 1 1 §)^i we may find the dependence r = r(a)o)
r = R exp
sin^ (p — "
(16)
Formula (16) determines the function r = r(a)o). I.e., the distribution /114
of density over the radius In implicit form, according to the specific
dependence of the beam exit angle on the angle of Incidence.
Performing measurements In different planes z = const, we may thus
obtain the density distribution along the cylinder axis in the case of
axial symmetry. For a fixed angle of incidence \l>, equation (15) determines
the beam exit angle <^q as a function of frequency. A knowledge of the
function <|)o('»)) enables us to determine the density distribution. We thus
assume
2(r) =
,(r)
E = arc sin I - sin ^j .
(17)
Equation (14) may then be written as
105
Illlllllllll II I
2W JYu>-^
Qi-
We thus find
' -' ' 2a. JdQ
Since 5(0) = ^io-n ~ 'l^» assvming that y = Q, -we can write
(as) du>
We thus obtain the expression determining the dependence n(r) in implicit
form
1^ p ?o H da> I (18)
-f + i|iJ^.)
Temperature Determination from Beam Damping
Due to collisions , the energy of a beam leaving the plasma is
eT times less than the incident energy, where t = / x ds is the
optical plasma thickness; x — damping coefficient. If the wave fre- /115
quency is considerably greater than the frequency of collisions of elec-
trons with ions and neutral particles v (r) , we then have
x(r) =
•■l/--¥
Knowing x(r), we may find the plasma temperature distribution from the
specific dependence of v on temperature.
The damping t depends on the angle of incidence f. We may find the de-
pendence of the damping coefficient on the radius from the quantity t(ijj),
which is measured experimentally, and from the specific density distribution.
Let us first investigate the planar problem (in the case of a cylin-
drical plasma, the beam trajectories lie in the plane ((> = const). For
106
T (ip) , we may write the following expression
dz
Substituting -J— from formula (5) In this, we obtain
x(.j))=2 1 x(a:)i/ "7"" , dx==2 \ X V '~" du. (19)
Let us Introduce the function f(u) according to the equation
(20)
We then have
xr<];)-2 ^"
J ]/ 00524/ — u
w = ^Ul/ i-;r»/\ i/io_
(21)
Solving this equation and equation (8) , we find
u
2
Assuming that A = 2» we obtain jWb
2\ p _a^^il^'l_ (22)
-2 — cos* 4'
..2 '
In explicit form, this formula determines the dependence of the damping
coefficient on coordinate x.
If the beam trajectories lie in the z = const plane, we then have
R
t(^)=2j x(r)|/"l + r^(g)'dr.
107
Substituting the value of -^ from formula (13) , we obtain
^(<1>)=2\ x(0
„2_<«2_<„2^sin*4-
dr.
(23)
Assuming that
„ '■' (a "A df(u) / V r 1 /" , »o dr
we may write equation (23) in the following form
1
J Yu — sin'
sin' 4«
(24)
Solving it in the same way as equation (19) , we find
r-^l—dsin'O/
Assuming that y = u(r), we obtain the following expression
:(r) =
dVu \
<^ J
di
dY u \ ds'\ni/
dsin<j/
dr J Ysm^if — u
/117
(25)
Expression (25) determines the dependence of the damping coefficient
coefficient on the radius according to the specific dependence of x on i|;.
REFERENCES
1. Landau, L. D. , Lif shits. Ye. M^ Mechanics (Mekhanika) . Fizmatgiz,
Moscow, 1958.
2. Firsov, 0. B. Zhumal Eksperimental'noy i Teoreticheskoy Fiziki, 24,
279, 1953.
3. Rydbeck, 0. Phil. Mag., 30, 282, 1940; 34, 130, 1943.
108
4. Ginzburg, V. L. Propagation of Electromagnetic Waves In a Plasma
(Rasprostranenlye elektromagnltnykh voln v plazme) . Fizmatgiz,
Moscow, 1960.
5. Smoys, S. Joum. of Appl. Phys., 32, 689, 1961.
109
&
SECTION III
PLASMA NONLINEAR OSCILLATIONS AND WAVE INTERACTION /118
EXCHANGE OF ENERGY BETWEEN HIGH FREQUENCY AND LOW FREQUENCY
OSCILLATIONS IN A PLASMA
V. D. Fedorchenko, V. I. Muratov, B. N. Rutkevlch
Oscillations with frequencies v on the order of 100 kc (Ref. 1),
as well as oscillations at the electron plasma frequency (a3oe ^ 50 Mc) ,
occur in a plasma produced by an electron bundle in a longitudinal magnetic
field at a pressure of 10-^ — 10"^ mm Hg. It has been found that there
is a relationship between the low frequency and high frequency oscillations
(Ref. 1-3). High frequency oscillations cause an increase in the low
frequency oscillation amplitude. On the other hand, they undergo ampli-
tude modulation at a low frequency, which leads to the occurrence of com-
bined frequencies w + v. The amplitudes of oscillations having the fre-
quencies 0) + V and o) - v are not the same, which — just as with the
Landsberg-Mandel ' shten-Raman effect — can indicate the direction and
effectiveness of energy transfer from oscillations of one frequency to
oscillations of another frequency.
As has been shown previously (Ref. 1), low frequency oscillations /119
represent ion motion (which is transverse with respect to the bundle) in
the field of the space charge of electrons contained by a magnetic field.
The frequency of these oscillations may be readily computed from the
following simple model. Let us assume that electrons and ions fill one
and the same cylindrical region in the equilibrium state. The simultaneous
shift of all ions by the quantity r^, and of electrons by the quantity r^,
leads to polarization, which can be determined as follows in the case of
a small shift
(1)
where n is the plasma density.
We can write the following expression for electrons and ions
110
I ■ III! ■ I
or, with allowance for (1) ,
r; = — (Boi {n — re) + — [nB];
nil
r, = o)o« (r/ — /-,) — — [r^B],
"'t
(4)
(5)
where
ne-
..2
"' 'o""' (6)
0«
i2
e„m,
o'"t (7)
Disregarding the electron shift during orbital motion, we shall only
allow for their drift In crossed electric and magnetic fields , and
we shall designate the position of the electron guiding center by the
vector Tg. According to expression (5), we then have
e r r., e
ri — r,=-- J- ['■'^1 = r f''«'^l- (8)
/n,-<o,
i"'Oi
It follows from formulas (4) and (8) /120
e
^'=-^f^'^J+i-.f^5I.'
and thus
^' = - ^ t^'^J + h. l^^l = -^ a^-S] B\ =
e>B'
i^oi ' '«,«>o/ ' (9)
or
(10)
where
2//=-^' (11)
"o<
— the ratio between the ion cyclotron frequency and the plasma frequency.
Substituting the expression obtained for r^ in formula (4) , we obtain
n-^CnB] + <o„^,(l + V)^ =0. (12)
Equation (12) has a r-^ 'V' e type of solution, and the frequency
111
must satisfy the following equation
Q^ + QhQ~1— Q-2 = 0.
I.e. ,
2i2 = -|-^±l/ ^f + I+V
(13)
(14)
In the case of f^g ^ 1, v ^ oiQi "^ •%/ "T*
m
X
If ^TT >> 1, equation (13) has two roots;
Qi = — Q„ (15)
22 =J-
Q^- (16)
If % << 1 Q=±i'.
(17)
The oscillation frequency v decreases with an increase in the magnetic
field strength, which coincides with the solutions of (16) and (17), if
it is assumed that the frequency oJq. depends slightly on the magnetic field
strength. If cOq. does not depend on B — i.e., Wq. is constant — it may /l^l
be readily determined according to any pair of measured v and B, which
would then enable us to compile a graph showing the dependence of v on B
according to formula (14) . The dependence of ion oscillation frequency on
the magnetic field strength is shown in Figure 1. The curves were re-
corded for different current values in an electron bundle (1 - I = 10 ma,
2 - I = 30 ma) . The dashed curves are drawn through the points computed
according to foirmula (14) , under the assumption that the plasma density
does not change when there is a change in the magnetic field strength.
The computed points do not lie on the experimental curves (particularly
in the region of small fields) , which is no doubt related to the variability
of (J^oH • Nevertheless, it may be stated that the nature of the dependence
is correctly imparted by our simple model.
Modulation of high frequency oscillations may be due to low frequency
oscillations of the plasma density. Let us assume, for example, that the
plasma is located in an external field Ee-"-*^*-, which is transverse with
respect to the bundle, and the frequency w is so great that only allowance
for the electron component shift has any meaning. The external field
ggiojt gives rise to a shift and the occurrence of a polarized field
112
'\< — 'r„ . The equation for electron motion in these fields may be written
in the following form
^~—ip-'+f-)-il'-S>- (18,
We obtain the following expression for forced electron oscillations
e pAoit
--1-Ee
+ »(">//« — "^
(19)
If it is assxjmed that the plasma density undergoes oscillations at the fre- /122
2 2
quency v << w and the quantity Ugg has the form tOgg(l + a cos vt) , while
a << 1, then expression (19) changes into the sum of the oscillations with
the principal frequencies (oi) and the combined frequencies (to + v) :
ete , a e t e
"^ 2 ■ -/ eg.
-; /((u—v) <
e £ e
(20)
The oscillations have resonance close to the electron cyclotron frequency.
This model is inadequate for determining oscillation intensity at the
combined frequencies.
The experiments were perfomaed on a hollow electron bundle in a longi-
tudinal magnetic field with a strength ranging between 200 - 2000 oersted.
The bundle length was 50cm; diameter — 2 cm; energy — 250 v; and the
current — 20 - 40 ma. The bundle was located in a metallic tube having
a diameter of 9 cm. Pressure in the chamber was several units of 10~° mm Hg.
The interaction between the outer, high frequency field and the low fre-
quency ion oscillations was studied. The interaction was observed in
three cases: Vlhen the frequency of the outer signal (1) did not coincide
with any of the plasma eigen frequencies, (2) coincided with the electron
cyclotron frequency, and (3) coincided with the plasma electron frequency.
In the nonresonance case, the outer field was transverse with respect
to the bundle. It was produced between two conductors having a diameter
of 0.2 cm and a length which was close to the bundle length. The conductors
were parallel to the bundle axis at a distance of 6.5 cm from each other.
Figure 2 shows the diagram of the experimental apparatus employed to /123
113
00400 aoeoo 0.1200 w."'"
Figure 1
study the interaction between ion oscillations and the outer electric
field, whose frequency does not coincide with any of the plasma eigen fre-
quencies (1 — solenoid; 2 — electron gun; 3 — probe; 4 — hollow elec-
tron bundle; 5 — collector; 6 — conductors between which a high fre-
quency field is produced; 7 — coils connecting the generator and the
measurement circuit. The resonance curve for the circuit producing the
high frequency field is shown in the upper right) . The capacitor is a
section of the resonance circuit weakly connected to the GS-23 generator
and the C4-8 spectrum analyzer. The resonance frequency of the circuit
was 13.88 Mc. The resonance barely shifted when the bundle was switched
on. The resonance width was quite large (Q = 50), so that the combined
frequencies did not go beyond it. The connection between the circuit, the
generator, and the measurement circuit was selected so that the resonance
curve was symmetrical. This was important for comparing the intensities
of combined oscillations with the frequencies cj + v and 03 - v.
Figure 3 presents typical spectra of oscillations produced when the
resonance frequency was used (the spectra were obtained under the following
conditions: Pressure p = 3.4'10~^ mm Hg, electron bundle current I = 30 ma,
anode voltage U =250 v, effective variable outer field strength v^ = 64 v,
magnetic field strength H: a — 600 oersted; b — 780 oersted; c — 920
oersted) . Lateral frequencies spaced at the frequency of ion oscillations
may be seen, in addition to the frequency employed. Employing the termin-
ology used in the theory of combined scattering, we shall call the lateral /124
lines red (o) - v) and violet (u + v) "companions". The relative height
of the companions (with respect to the carrier height) was 1-2%, and it
increased with an increase in the amplitude of ion oscillations and the
amplitude of the outer signal.
The heights of the red and violet companions, generally speaking,
were different, and this difference depended on the magnetic field
114
rt-M-e
TW^ KC^-a
Figure 2
strength. As may be seen from Figure 3, when the magnetic field strength
was 600 oersteds, the violet companion was higher, and for 920 oersteds
the red companion was higher. At a field strength of 780 oersteds, the
companions were the same. In order to explain the difference in the
heights, we should point out that the increase in the ion oscillation
amplitude when a high frequency outer signal was employed was always
greater for large magnetic fields. For small magnetic fields and the
same intensity of the outer signal, the amplitude increase in ion oscilla-
tions disappeared. Thus, an increase in the magnetic field strength can
improve the conditions for transferring energy from high frequency oscilla-
tions to ion oscillations.
Comparing this with the data given in Figure 3, we may arrive at the
conclusion that there is a relationship between the companion heights and
the direction of energy transfer. The predominance of the red companion
corresponds to the transfer of energy from high frequency oscillations to
low frequency oscillations; the predominance of the violet companion
corresponds to the energy transfer in the opposite direction. The direc-
tion of energy transfer no doubt depends on the relationship between the
amplitudes of the ion oscillations and the outer signal. In actuality,
an increase in the magnetic field strength decreases the ion oscillation
amplitude, which leads to a more effective energy transfer from the high
frequency to the low frequency for a given amplitude of the outer signal.
The same result may be achieved in another way: by changing the
outer signal amplitude for a constant magnetic field. With an increase
in the outer signal, the red companion becomes higher as compared with
the violet .
/125
Let us turn to an experiment in which the outer field frequency
coincides with the electron cyclotron frequency. Figure 4 shows the
115
Figure 3
diagram of the experimental apparatus for studying the interaction be-
tween ion oscillations and the outer field at the electron cyclotron fre-
quency (1 — solenoid; 2 — electron gun; 3 — probe; 4 — hollow electron
bundle; 5 — collector; 6 — volumetric resonator; 7 — connection with
the oscillation source and measurement circuit) . The outer field was
produced in the volumetric resonator which encompassed almost all of the
bundle. The mode Hj^, was excited in the resonator at a frequency of
2265 Mc. When the bundle was switched on, the oscillation level in the
resonator sharply decreased, when the magnetic field strength reached a
value corresponding to electron cyclotron resonance (Figure 5) . The
width of the resonance absorption curve was primarily determined by the
116
J— KS4-5
/(gs-22
Figure 4
A.rel
lunit
QOBOO 0.1200 H,mJ!,
Figure 5
nonuniformlty of the magnetic field over the length of the system. Compan-
ions appeared close to resonance which coincided with formula (20) . The
violet companion was higher than the red companion, which was apparently
related to the small amplitude of the signal supplied.
Let us examine the case when the outer field frequency coincides
with the electron plasma frequency. The outer signal is supplied to a
grid located at the bundle origin. Measurements are performed by a probe.
Figure 6 shows the spectra for several frequencies close to the electron
plasma frequency (p = 4-10-^ mm Hg, I = 40 ma, U = 250 v, H = 1000 oersted,
voltage on the grid u^ = 0.1 v. Oscillation frequencies of voltage on the
grid: a — 39 Mc; b — 40 Mc; c — 41 Mc; d — 42 Mc; e — 43 Mc) . It
can be seen that the interaction is resonant in nature. The heights of
the companions are large (the total altitude of the carrier is shown in
the photographs) , which points to the effectiveness of the interaction be-
tween ion oscillations and the outer signal at the electron plasma fre- /126
quency .
The amplitude of the signal supplied is 0.1 v , which explains the
* The effective voltages are always employed.
117
Figure 6
predominance of the violet companion. When the amplitude of the outer
high frequency field increases (for a magnetic field strength of 1000
oersted), the red companion begins to predominate. A similar relationship
between the companion heights and the amplitude of the outer field is ob-
served for H = 7000 oersted. With an increase in the red companion, the
altitude of the principal line (carrier) decreases. If H = 700 oersted,
a decrease in even the principal line may be observed when the outer
signal is intensified (from u^. = 0.5 v to u^ = 0.95 v) . For a compara-
tively small magnetic field (H = 400 oersted) and a significant outer
signal amplitude, there is very strong interaction which is accompanied
by the appearance of many combined frequencies .
These data point to the effective transfer of energy from electron
118
plasma oscillations to Ion oscillations. This phenomenon may probably
be employed to increase the energy of the plasma ion component.
REFERENCES
1. Fedorchenko , V. D. , Rutkevich, B. N. , Muratov, V. 1., Chernyy, B. M.
Zhurnal Teoreticheskoy Flziki, 32, 958, 1962.
2. Fedorchenko, V. D. , Muratov, V. I., Rutkevich, B. N. Zhurnal Teoreti-
cheskoy Fizikl, 34, 458, 1964.
3. Fedorchenko, V. D. , Muratov, V. I., Rutkevich, B. N. Zhurnal Teoreti-
cheskoy Fiziki, 34, 463, 1964.
DISSIPATION OF PLASMA OSCILLATIONS EXCITED IN A
CURRENT-CARRYING PLASMA
Ye. A. Sukhomlin, V. A. Suprunenko, N. I. Reva, V. T. Tolok
Several experimental and theoretical investigations have studied the
development of bunched instabilities in a current- carrying plasma for
large electric field strengths (Ref. 1-5). It has been shown that, as
only the mean energy of the ordered electron drift is larger than their
thermal energy, intense longitudinal electrostatic oscillations develop /127
in a plasma. Their energy reaches the initial energy level of electron
drift usually after several tens of plasma oscillation periods.
Computations of the multi-flux motion of electrons in a current-
carrying plasma, which were performed by 0. Buneman (Ref. 6), J. Dawson
(Ref. 7), and Ya. B. Faynberg (Ref. 8), have shown that very intense
"thermalization" of the plasma oscillation energy occurs, if this energy
is considerably greater than the electron thermal energy. Thermalization
occurs due to nonlinear phenomena leading to the transformation of longi-
tudinal oscillations into transverse oscillations and to their rapid phase
mixing .
This process takes place until the energy of the ordered oscillations
equals the electron thermal energy. It may be assumed that ion heating
119
will occur due to "collective" friction of electrons on ions in the case
of bunched instabilities.
Thus, an investigation of bunched instabilities in a high-current
gas discharge opens up new possibilities for effective plasma heating.
The studies (Ref. 9, 10) performed detailed investigations of the
excitation conditions of bunched instabilities in a current-carrying
plasma, as well as the plasma characteristics in the presence of these
instabilities. The occurrence of an anomalously high discharge resistance
and intense microwave plasma radiation was discovered.
This article investigates heating and containment of a plasma in a
strong magnetic field, under conditions when bunched instabilities excited
by "escaping" electrons develop in the plasma. The experiments were per-
formed on an apparatus representing a rectilinear tube made of alundvim
with a diameter of 10 cm and a length of 25 cm, which was usually filled
with hydrogen at a pressure of 5'10~^ - lO"** mm Hg. Aluminum electrodes
were placed at the two ends of the tube; a battery of capacitors having
an over-all capacitance of 15 microfarads was discharged between the
electrodes. The battery was charged to a voltage of 30-40 kv. The
discharge current through the gas amounted to 100 ka with a period of
9 microseconds. In order to eliminate hydromagnetic phenomena, the dis-
charge was performed in a strong longitudinal magnetic field (on the order
of 1.2 tl) , at which the Shafranov condition of stability would be ful-
filled (Ref. 13). In order that the plasma did not touch the walls, a
diaphragm with an opening which was 80 mm in diameter was placed between
the electrodes.
During the first half-period in the discharge, a highly ionized plasm a/128
filament, which was separated from the wall and which had a diameter of
80 mm, was produced; no macroscale hydromagnetic Instabilities were
apparent in this plasma filament. The plasma density changed between
10 - 10^^ cm~^. The construction of the apparatus and the experimental
method were described in detail in (Ref. 9). X-ray and microwave radia-
tion from the discharge, the current of "escaping" electrons, the over-all
discharge current, and the voltage between the electrodes were studied
experimentally.
Figure 1 shows the following oscillograms : a — microwave radiation
from the plasma; b — over-all discharge current; c — current of "escaping"
electrons; d — voltage between the electrodes of the discharge tube re-
duced to a single time scale. The oscillograms were recorded at an initial
hydrogen pressure in the chamber of 2-10"^ mm Hg, a magnetic field strength
of 0.64 tl, and a charge voltage of 34 kv.
Characteristic, inter-correlated oscillations are observed at high
120
t jiaicrosec
Figure 1
electric field strengths ; these oscillations are due to the development
of bunched instabilities. During the initial period, the field strength
in the discharge center increases as the voltage wave penetrates the
plasma. With an increase in the electric field strength in the discharge,
accelerating processes begin to develop. A current of "escaping" electrons
first appears due to the "tail" of the Maxwell distribution. However, as soon
as the electric field strength in the plasma begins to exceed the criti-
cal value, all of the electrons acquire a drift velocity which is greater
than the thermal velocity, and bunched instabilities develop in the plasma,
to which the occurrence of epithermal microwave radiation corresponds.
In this case, a large portion of the directional drift energy of the
electrons is transmitted to excitation of oscillations, and the current
of "escaping" electrons sharply decreases. This leads to an increase /129
in the effective plasma resistance and to a dip on the oscillogram for
the over-all discharge current.
The amount of energy contributed by the outer source to the buildup
of longitudinal electron oscillations may be computed from the additional
current at the moment an instability develops. For the case shown in
Figure 1, this energy amounts to 10 kv per particle. The energy of these
oscillations considerably exceeds the initial thermal energy equalling
30 electron volts, which leads to effective thermalizatlon of plasma
oscillations due to nonlinear phenomena. As a result, the random electron
energy will equal, in order of magnitude, the energy of plasma oscilla-
tions. Due to thermalizatlon, intense X-ray radiation occurs as a result
of energetic electrons falling on the target.
121
.£■
k
^
3
0.5
\
nT^"^
I.
\.
■^v^^^
^
V ^"^
7—- s>
^^
1
2
n
1 /
0,2 Ofi S.HM
Figure 2
Since the transverse component of electron energy has increased con-
siderably, and the velocity of directed drift has remained the same as
previously, the condition for excitation of plasma bunched instabilities
has been disturbed and the plasma has returned to the initial unexcited
state. Figure 1 clearly illustrates two cycles of such oscillations with
a period on the order of 1.5 microseconds. This period is probably deter-
mined by the time required for electrons to change into a state of "escape".
According to the computations of Dreicer (Ref. 11), this period equals,
in order of magnitude, the time between two Coulomb collisions for an
electric field strength which is greater than the critical strength.
Assuming that 30% of the energy of plasma oscillations is "thermalized"
(Tg = 3*10^ electronvolts) (Ref. 12), we find that for a plasma density
of 7 '10-^^ cm~^ the time between two Coulomb collisions is 4.5 microseconds.
The period in which the "heating" cycles are repeated depends on the
initial gas pressure in the chamber. It increases considerably with a
decrease in the plasma density. The effective electron temperature of the
plasma must thus increase, since the total energy transmitted into the
buildup of plasma oscillations from the outer source changes very little.
Thus, it would be expected that intense electron heating occurs due
to the development of bunched instabilities in the discharge. Direct
measurements of the electron temperature are of great interest.
The effective electron temperature was determined by the absorption
of electron braking radiation in thin beryllium foils located in front
of a scintillation crystal on the wall within the vacuum chamber. The
plasma electrons falling on the foil-target are braked in the very thin
surface layer. Their energy is transformed into braking X-ray radiation
which, after partial absorption in the foil, falls on the crystal causing
a flash of light. In order to determine the radiation hardness, without
disturbing the vacuum it is possible to place beryllium foils having
different thicknesses in front of the crystal. The light from the crystal
/130
122
Figure 3
is supplied to the photomultlpller by means of the wave guide. The signal
from the photomultlpller through the cathode follower is supplied to the
oscillograph amplifier.
Figure 2 shows the dependence of the photomultlpller (PM) signal
intensity on the thickness of the absorber foil. This dependence may
be employed to determine the electron temperature for a specific form
of the electron energy distribution function. Maxwell distribution,
rectangular distribution with the width T^, and Drayvesten distribution
lead to similar temperature values . This enables us to employ the curve
shown in Figure 2 for a rough estimate of the electron temperature when
the electron energy distribution function is not known precisely.
The curves in Figure 2 were compiled under the assumption of Maxwell
distribution for three temperatures: 1 — 1 kev; 2 — 2 kev; 3-3 kev.
It can be seen that the experimental points correspond to a plasma elec-
tron temperature on the order to 2 kev.
Figure 3 presents the following oscillograms : a — current of
"escaping" electrons ; b — X-ray radiation from the plasma due to braking
123
J f ' I I L_i 1 1. .*.-J _
0^10
mxcrosec
Figure 4
of thermalized electrons in the foil. X-ray radiation arises simultaneously
with the current , and continues for a long period of time after the
braking of "escaping" electrons due to the energy of transverse motion.
The maximum quanta energy of this radiation is about 15 kev. This points
to the effective transformation of the energy of electron longitudinal os-
cillations into the energy of transverse motion. In the absence of
bunched instabilities, the electron temperature in the discharge is 30
electronvolts .
124
Figure 4 presents the following oscillograms: a — current of /131
"escaping" electrons; b — X-ray radiation; c — microwave radiation;
d — light from the discharge center; and e — over-all discharge current.
X-ray radiation arises simultaneously with intense epithermal microwave
radiation at a frequency close to
y<+<-]/"^.
(wq — plasma electron frequency; tOg — electron cyclotron frequency).
The power of this radiation is approximately four orders of magnitude
greater than the power of thermal radiation from the plasma at an elec-
tron temperature of 10** electronvolts. It was shown in (Ref. 10) that
the radiation is related to longitudinal electron oscillations in the
discharge.
All of these statements indicate that X-ray radiation from discharge /132
is related to the thennalization of bunched instability energy.
The period of pair interaction for electrons having an energy on
the order of 2 kev is considerably greater than the time of the process
being studied. Therefore, we may assume that nonlinear phenomena during
collective oscillations with a large amplitude play the main role in
thermalization of longitudinal oscillations. In our case, the Larmor
radius of hot electrons is a little less than the discharge chamber
diameter (less than 1 mm). Therefore, X-ray radiation is primarily
caused by electrons diffusing toward the walls across the magnetic field.
When the X-ray radiation reaches a maximum (see Figure 4) , the lumines-
cence in the discharge center sharply increases; this is determined pri-
marily by admixtures dislodged from the walls by hot electrons. There is
a simultaneous strong increase in the current on the boundary electro-
static probe, which is executed in such a way that the current upon it is
determined by the resistance of the discharge plasma across the magnetic
field. Thus, there is a rapid cooling of the electrons, even 5-6 micro-
seconds after the X-ray radiation has terminated.
In the absence of pair collisions, strong diffusion and great con- /133
ductivity across the magnetic field would not be expected. Anomalous
diffusion can occur due to nonlinear phenomena for a large longitudinal
oscillation amplitude. However, in the experiments described, the anoma-
lous diffusion across the magnetic field may be explained by an increase
in the gaskinetic plasma pressure as compared with the magnetic pressure,
due to intense electron heating. The decrease in the X-ray radiation in-
tensity with an increase in the magnetic field strength also points to
this conclusion.
125
REFERENCES
1. Budker, G. I. In the Book: Material From the Conference on the
Peaceful Utilization of Atomic Energy (Materialy konferentsii po
mirnomu ispol'zovaniyu atomnoy energii) . Geneva, 4, 76, 1956.
2. Thomassen, K. Phys. Rev. Letters, 10, 80, 1963.
3. Shepherd, L. J., Skarsgard, H. M. Phys. Rev. Letters, 10, 4, 121-
123, 1963.
4. Zavoyskiy, Ye. K. Atomnaya Energiya, 14, 57, 1963.
5. Fanchenko, S. D. , Demidov, B. A., Yelagin, N. I., Ryutov, D. D.
Zhurnal Eksperimental'noy i Teoreticheskoy Fiziki, 46, 497 - 500,
1964.
6. Buneman, 0. Phys. Rev., 115, 503, 1959.
7. Dawson, J. Phys. Ref., 113, 383, 1959.
8. Faynberg, Ya. B. Atomnaya Energiya, 11, 313, 1961.
9. Suprunenko, V. A., Faynberg, Ya. B., Tolok, V. T., Sukhomlin, Ye. A.,
Reva, N. I., Burchenko, P. Ya. , Rudnev, N. I., Volkov, Ye. D,
Atomnaya Energiya, 14, 349, 1963.
10. Suprunenko, V. A., Sukhomlin, Ye. A., Reva, N. I. Atomnaya Energiya,
17, 83, 1964.
11. Dreicer, H. Phys. Rev., 115, 238, 1959.
12. Shapiro, V. D. Zhurnal Eksperimental'noy i Teoreticheskoy Fiziki,
44, 613, 1963.
13. Shafranov, V. D. Atomnaya Energiya, 5, 38, 1956.
126
■■ III mil I mil HHHi mil i i ii i i n
I
DAMPING OF INITIAL PERTURBATION AND STEADY FLUCTUATIONS
IN A COLLISIONLESS PLASMA
A. I. Akhiyezer, I. A. Akhiyezer, R. V. Polovin
When the perturbation of the distribution ftinction is a nonanalyti-
cal function of velocity, the damping of plasma waves when there are no
collisions may be different from that determined by the classical formula
of Landau. If the perturbation of the distribution function has a /134
6-like singularity, then perturbations of the macroscopic quantities are
not damped with the passage of time. If the perturbation of the distribu-
tion function has a discontinuity of the n-th derivative, then the perturba-
tion is damped according to the law t-(n+l) (and not according to the
exponential law, as in the case of Landau damping).
Let us investigate the mechanism for establishing fluctuations
(which are not dependent on the initial perturbations) of the macroscopic
quantities in a non-equilibrium plasma. We shall show that these fluctua-
tions are established due to the "survival" of a singular component in
the expression for the distribution function perturbation.
The dependence on time of the perturbation of the k-th component of
the Fourier potential <t>i^(t) in an unconfined plasma is determined by the
following expression (Ref. 1)
f+'^o (1)
O — ioo
'^^^^^ _ 4^ Af (k, p) .
?Ap - k^ ' D (k, p)'
(2)
N{k,p)=jjk{w)~^; (^j
w=^ ,F^iw) = ^Fo{v)dvj., gk(w)^^gki\)dv^, vx=v--k|;
Fo(v) is the unperturbed distribution function; g^Cv) — the Fourier com-
ponent of the distribution function perturbation in the case of t = 0;
integration is performed in formula (1) along the line Re p = a lying to
the right of all the singularities of the function i> .
Formula (1) makes it possible to determine the behavior of the poten-
tial <f>]j^(t) with an increase in t. As is known, the asymptotic behavior
127
of the function <i)]j_(t) for large t is determined by the nature of the
singularities of the Laplace transform (J)^. The function (|)p was
detemnined above for only large values of Re p (in this region it has
no singularities). In order to study its singularities, we must first
determine this function in the entire complex variable plane — i.e.,
we must analytically continue the determination of (2) to decreasing
values of Re p. The analytical continuation of (jip is determined according
to the previous formula (2) along the imaginary axis p.
For purely imaginary values of p, the denominators in the integrals /135
which determine (j)„ vanish in the case of w = ip/k. Therefore, for
analytical continuation of (j) in the region Re p ^ 0, it is necessary
to deform the integration path in the integrals (3) and (4) , so that it
passes around the pole w = ip/k from below. Deformation of the path
assumes, in its turn, the possibility of analytical continuation of the
functions F^ (w) and gjj^(w) determined initially only for real w in the
region of complex values of w.
Thus, a clarification of the singularitites for the function ((Jcr*
which determines the nature of the asymptotic behavior of i|)]j^(t) for large
t, requires a knowledge of the analytical properties of the functions
Fo(w) and g^Cw).
Let us confine ourselves to investigating the functions Fq (w) per-
mitting analytical continuation in the region of complex values. The
function D(k, p) , determined in the case of Re p > by relationship (3),
may be continued analytically in the region Re p .< 0, by determining it
everywhere as
'x^.")"'-^^!!!^' «
-0' +
where integration is performed along the real axis w with passage around
the pole from below in the case of w = ip/k.
We have found the denominator of expression (2) for * — i.e., the
function D(k, p) — over the entire plane of the complex variable p. Let
us now calculate the analytical properties of the numerator for this expres-
sion, i.e., the function N(k, p) . Formula (4) determines it in the case
of Re p > 0. As has been pointed out, the fimction N(k, p) has no singu-
larities in this region. The position and nature of the singularities of
this function are determined by the properties of the fvinction gjj^(w) in
the case of Re p ^ 0.
If the function gj^(w) has singularities (which may be integrated) for
128
real w, then the function N(k, p) will have singularities for purely
imaginary p. In particular, such a situation is observed if the func-
tion g]5^(w) has a 6-like singularity, a discontinuity, or a break, and
also if any of its derivatives has a break (in these cases, the func-
tion gjj^Cw), generally speaking, does not permit analytical continuation
on the real axis.
If the function g]^(w) has no singularities on the real axis and per-
mits analytical continuation in the region of complex values of w, then
the function N(k, p) , and consequently the function (j) , will have no
singularities on the imaginary axis p. However, generally speaking, it /136
may have a singularity in the case of Re p < at the points p = -ikw^.,
where Wj. is any singularity of the function g^^Cw) lying in the lower
halfplane of the complex variable w.
Let us elaborate further on the nature of the asymptotic behavior of
(t)jj^(t) for noninteger ftmctions g^(w) . In this case, singularities of the
function N(k, p) are added to the singularity ^ determining the roots of
the dispersion equation D(k, p) = 0. The distribution of these singulari-
ties depends only on the form of the function gu(w) — i.e., on the nature
of the Initial perturbation — and does not depend on the plasma properties
(on the function Fg (w) . ) As has been indicated, one significant property
of the singularities for the function of N is the fact that they may all
lie only in the left halfplane p. Therefore, if only one of the roots
Pj. = -ia)j,-Yr of the dispersion equation D(k, p) = lies in the right
halfplane p, Yr *^ (which corresponds to the possibility of an oscilla-
tion increase) , then the nature of the initial perturbation has no signifi-
cant influence on the asymptotic behavior of <l'i^(t) in the case t t ->■ °°.
If N(k, p) has singularities at the points p = p^^ = -Yn~i<J^n (ii = 1>
2, 3, ...), the contribution made by these singularities to the asymptotic
behavior of <t'k(t) in the case of t ->- «> may be written as2j""®^P{ — T«^ — "o«0,
where a.^ represents certain constants. Adding this svim to the contribu-
tion from zeros D(k, p) , we find the asjnnptotic expression for 'f']j^(t) in
the general case of noninteger functions g]j^(w) (which have no singulari-
ties for real w)
"fx (0 ~ S?!'-^ exp { —i^i - /CO,/} + ^ a„ exp {—^J - /co„/}, ^^^
where <\>^^^ is the residue (Ji^,^ at zero of the function D(k, p) (the point
cr
P = Pr = -Yr-if^r^*
129
Thus, for large t the potential <i>v(t) represents superposition of
the eigen plasma oscillations, whose complex frequecies a)j.-iY^ are deter-
mined by the plasma properties [the right sum in (6) ] , and oscillations
whose complex frequencies (jJ^-iY are determined by the form of the initial
perturbation gi^(w) [the second sum in (6)], The eigen oscillations may
be both damped and intensified. The oscillations whose frequencies are
determined by the form of the function gi^(w) may be only nonincreasing
(i.e., damped or oscillating oscillations with constant amplitude).
Let us give two examples of oscillations whose frequency and damping
decrement are determined by the initial perturbation, and do not depend
on the plasma properties.
As the first example, let us investigate oscillations produced in /137
the case of
gkiw)
go^i
(w-w,y+w^i' O)
where gg , wq , wj are certain constants. In this case we have
N (k, p) = , ■■ "\ . .
The function N(k, p) has a singularity in the case of p = -ikwQ-kwi,
which introduces the following contribution to the asymptotic behavior
of <l'v(t) in the case of t -^ ~
fk (0 "^ ^0 exp [—kwif — ikw^t]. (8)
Thus, the frequency and damping decrement of oscillations produced in
the case of initial perturbation such as (7) equal kwQ and kwj_, respec-
tively. In the case of wj -^ 0, the damping disappears. We should point
out that the function gi^(w) acquires an 6-like singularity on the real
axis, g]^(w) -^ Trgo6(w-wo).
Let us study oscillations produced in the case of the discontinuous
functions g, (w). Let us set, for example,
^0 (—^0 <W< Wo), (9)
Thus, the function
^^(^)-lO {\w\>w,)
130
^ '^' IK p — ikW
has branch points on the Imaginary axis p, p = + ikw. The contribu-
tion made by the singularities of the function N(k, p) to the perturba-
tion of the potential (("kCt) has the following form
^,(0~^/iiii«. (10)
The 6-like singularity on the real axis on the function g}^(w) leads to
nondamped oscillations of the potential 'l'ir(t) • The discontinuity of the
function gj^Cw) — i.e., the 6-like singularity of its first derivative —
leads to potential oscillations which are damped as t~^ . It can readily
be shown that the discontinuity of the n-th derivative of the function
g]j^(w) — i.e., the 6-like singularity of its (n + l)-th derivative —
leads to asymptotic behavior of a t~(^'''-'-)exp{ikwot} like potential, where
wg is the discontinuity point.
Let us determine the manner in which the fluctuations of macroscopic /138
quantities, which do not depend on the initial conditions, are established
in a collisionless plasma with an arbitrary (not necessarily equilibrium)
distribution function (Ref. 2). In order to do this, we should note that
the averaged product of the distribution function fluctuations for particles
of the a-th type can be represented as follows at corresponding periods
of time
< 8t (V) gt- (V) > = iaa'^ (k + k') 8 (V - V') fS (V) -f Yaa' (v, v'; k, k'). ^^^^
where the first component (Ref. 3) describes the "correlation of the particle
with itself" , and the second component is related to the interaction between
particles (and is determined by the previous history of the system) . It
is important that the first component contain 6(v - v'), while the second
component is a smooth function of velocity.
In order to obtain the correlation function of the potential, we
must express the potential <t']j(t) by gg(v) by means of relationships (1) -
(4) , and we must then perform averaging by means of formula (11) . It may
be readily seen that the presence of the 6-like component in formula (11)
leads to a nondamped (and oscillating according to the law expdt^yfc}) com-
ponent in the expression for the potential correlator. The remaining
(smooth) components in formula (11) are damped according to the law
exp{-Yi^t}, where Yi, is the customary Landau damping decrement (i.e., the
imaginary part of the root ny, of the dispersion equation e(a), , k) =0).
131
Thus, in the time yy~ the potential fluctuation distribution (and
the distribution of all other macroscopic quantities) , which does not
depend on the initial perturbation, is established after the outer per-
turbation is shut off in the plasma.
According to relationships (1) - (4) , (11) , the correlation function
of the potential has the following form
-< n (0 <PA'(0) > = 8 (k + ki^Y r do) I s (to.k) |-2 exp { - mt) x
^ -i (12)
X2e2^dv/=?(v)8(u>-kv).
a
This relationship was obtained by Rostoker (Ref . 4) by employing a
different method. Our derivation presents a clearer explanation of the
mechanism, and makes it possible to determine the time required to estab-
lish fluctuations, which do not depend on the initial perturbation, in a
nonequilibrium plasma.
REFERENCES
1. Landau, L. D. Zhumal Eksperimental'noy i Teoreticheskoy Fizikl, 16, /139
574, 1946.
2. Akhiyezer, I. A. Zhumal Eksperimental'noy i Teoreticheskoy Fiziki,
42, 584, 1962.
3. Kadomtsev, B. B. Zhumal Eksperimental'noy i Teoreticheskoy Fiziki,
32, 943, 1957.
4. Rostoker, N. Yademyy Sintez, 1, 101, 1961.
132
CHARGED PARTICLE INTERACTION WITH A TURBULENT PLASMA
I. A. Akhlyezer
This article computes the energy lost (or acquired) per unit of time
by a charged particle when moving through a turbulent plasma. The depen-
dence of the particle energy change on the magnitude and direction of its
velocity is established in explicit form. It is shown that the turbu-
lence spectrum elements have no influence on this dependence.
The strength of charged particle interaction with the plasma is deter-
mined by the level of plasma fluctuations. In particular, the particle
energy losses per unit of time P are related to the charge density corre-
lator <P^>qtQ by the following relationship
" = - '-f" I(?)'» < '' > '-« (•-''- "f ) fw • ">
where ez, y, v are, respectively, the particle charge, mass, and velocity.
If the plasma consists of cold ions and hot electrons moving at the mean
velocity u with respect to the ions, the charge density correlator in
the "sound region" (q(T^/M) ^2 << ^ << q(T /m) ^2^ aq << i) may be repre-
sented in the following form
( P' > ,. = f 7' {aqr {[t {q, qu) - ^ "-^(o] 8 (co - qs) +
+ [t {q, -qu) - i-'^u)] S (u, + gs)] ,
where T„, T- is the temperature and m, M — the mass, respectively, of
electrons and ions; s = (T„/M) ^^ — speed of sound; T(q, qu) — effective
1 1
temperature of sound waves; a = T ' 2(4Tre^n)~ '^ — Debye radius. Substi- /lAO
tuting (2) in (1) and assimiing that T >> pv^ , we obtain
(eza)^us .
P =
' ^g'dq f dcp jcos 9 - (^ - l) '%in 6 cos f\§-^T {q, tj). (3)
where 6 is the angle between v and u
Tj^qu = ^u |-cos9 + (1 — ^1 sin9cos<p|.
In the wave vector region in which the sound waves are damped (qu <
< qs) , the effective temperature is (Ref. 1-3) Tg(l - qu/qs)~^. Close
to the boundary of the stability region (qu ^ qs) the function T increases
133
sharply. In the case of qu > qs, the linear theory predicts an exponen-
tial increase in T with time. This increase stops due to nonlinear phe-
nomena, and a stationary fluctuation distribution is established \rtiich
is characterized by a very high effective temperature, i.e., the state
of stationary turbulence (Ref. 4).
It follows from this behavior of the ftinction T(q,Ti) that 3T(q,n)/3n
has a sharp maximum for a certain value of n close to qs, n = no '^ qs.
Noting that the energy losses are determined by the derivative 8T/3n,
and not by the function T itself, we can thus express P by a small number
of parameters which characterize 9T/8n in the case of n ^ tio» without
Including more detailed properties of the turbulence spectrum.
Close to n '^ no» we have
{,^r(,..)}-. =,^pyM(.) + xi(,)(i -fj}, (4)
where \i 2 equals unity in order of magnitude, and T* is a large quantity
equalling T(q, n) in order of magnitude in the case of n > qs.
Substituting (4) in formula (3) and assuming that 6^ < 6 < e_, where
cos e± = (ay)-' {s^ ± (v^ — s^^'^w" — s^)'/.}, (5)
we obtain
P = ^-^S-' '^ ^^ (^°s ^+ - COS ^y^'' (COS - cos 9_)-'A, (6)
where a = afiaq)^ {\i\2)~^^'i- This relationship determines the dependence
of P on V in explicit form. In the case of 6 < 60(60 = arc cos s^/uv) ,
the particle energy decreases and in the case of 6 > 60 it increases.
In the case of |6 - 6^. ]^ Tg/T*, relationship (6) ceases to be valid. /ML.
In the case of | 6 - 6+ | << T^/T*, we have
-aV("fT,sin9^//' IV / V / • (7)
where a^ =^aliaq)^\T'^'\V'^'dq (the signs «+» correspond to 6 ;^ 6^ ) .
Thus, P is proportional to T* in the region 6, < 6 < 6_, except for the
boundary of this region where P "^ (T*) '2.
In the case of v -^ u, the critical value of the angle 6^. strives to
134
zero, and relationship (7) does not hold. For 6^. << T^/T*, and employing
formulas (3) , (4) , we obtain
In this case, the energy losses are particularly large (they are propor-
tional to (T*2) .
If < 6^. or 6 > e_, then the expression for the energy loss does
not contain the large parameter T*/Tg. For | - 6^ | « 1, nevertheless,
P is proportional to |6 - 6^. |~ /z, and consequently it is large.
In order to have critical values of the angles 0^ , it is necessary
that both u and v exceed s. If u ^ s , then 0. ^ = arc cos s/v. In
± "c
the case of ol 0c» the energy losses are propotional to (T*) /2
When expressions (6) - (9) were derived, it was assumed that the
difference v - s was not too small. If 1 - s/v << T^/T*, the energy
losses will be at a maximum in the case of vu = s^ (in this case P is
determined by formula (8) and sharply decreases with an increase in
|vu - s^l .
In conclusion, we would like to point out that the dependence of P
on the angle holds, even if the assumption that 9T/3n is small in the
region n > no is not fulfilled. The contribution made by the quantity
9T/8n with n > Ho in the expression for P can only change the function P
somewhat in the case of 6^ < < 6_, without changing P in the case of
0^0^. Consequently, the nature of the dependence of P on the angles
is not changed in the case of 0^ <: .< 0_. In particular, the function
P, which is positive in the case of = 0+ and negative in the case
of = 0_, must vanish for a certain value of the angle = Oq, 6+ <
< 0Q < 0_ (thus 00 can differ somewhat from arc cos s^/uv) .
REFERENCES /1A2
1. Ichimaru, S., Pines, D. , Rostoker, N. Phys. Rev. Letter., 8, 231,
1962.
2. Ichimaru, S. Ann. of Phys., 20, 78, 1962.
135
3. Bogdankevich , L. S., Rukhadze, A. A., Silin, V. P. Izvestlya Vuzov.
Radiofizika, 5, 1093, 1962.
4. Kadomtsev, B. B., Petvlashvlli, V. I. Zhurnal Eksperimental'noy i
Teoretlcheskoy Fiziki, 43, 2234, 1962.
THEORY OF NONLINEAR MOTIONS OF A NONEQUILIBRIUM PLASMA
I. A. Akhiyezer
As is well known, low frequency oscillations with a linear law of
dispersion — so-called ion sound — are possible in a collisionless
plasma consisting of hot electrons and cold ions (Ref. 1, 2). It is
interesting to study nonlinear motions of a collisionless plasma consisting
of hot electrons and cold ions, and primarily simple waves. The study of
simple waves not only makes it possible to trace the development of an
initial perturbation, but it is also of interest as an independent investi-
gation, since only the region of simple waves can (when there are no dis-
continuities) be contiguous to an unperturbed plasma [see (Ref. 3)].
A. A. Vedenov, Ye. P. Belikhov, and R. Z. Sagdeyev (Ref. 4) have
studied simple waves in a two-temperature plasma on the basis of an ios-
thermal hydrodynamic model. This article investigates simple waves in a
nonequilibrium plasma on the basis of a kinetic equation, without employing
a special model. A system of equations has been obtained for the moments
(introduced in a specific way) of the electron distribution function, which
enabled us to determine the direction of change for quantities characterizing
the plasma in a sound wave, and to trace the development of a perturbation
having finite amplitude*.
» Equations Describing a Simple Wave /143
The system of equations describing the motion of a collisionless
Yu. L. Klimontovich and V. P. Silin (Ref. 5) have investigated the
problem of a hydrodynamic description of a two-temperature plasma
without collisions. Nonlinear motions of such a plasma were studied
in (Ref. 4, 6).
136
plasma consisting of hot electrons and cold ions has the following form
(l + "l)" + :&E = 0; |^n + div(«u) = 0; (D
div E = iTze (^Fdv — n) ; rot E == 0,
where F(v) is the electron distribution function; n and u are, respectively,
the ion density and hydrodynamic velocity; E — electric field; m, M —
electron and ion masses, respectively. (It is assumed that the electron
mean energy considerably exceeds the ion mean energy.) Being interested
in sound oscillations whose phase velocity is small as compared with the
mean thermal velocity of electrons, we do not have to take the term 8F/3t
into account in the first of the equations (1) . Confining ourselves to
one-dimensional plasma motions and making allowance for the fact that the
charge spatial distribution is small in a soundwave
(X — the length at which the quantities characterizing the plasma change
significantly; a — Debye radius), we can reduce the system of equations
(1) to the following form
dF __M du dF _ f,. ^ . „ ^ _ n-
dx m ' dt ' v^dv^ ~ ' dt '^ dx — ^' (2)
n — ^ Fd\,
where d/dt = 8/9t + u^^x (the x axis is selected in the direction of wave
propagation; the subscript x for the velocity component u^^ is omitted from
this point on) .
Let us introduce the "moments of the distribution function":
D; (X. t) = (-2)' r ^ F (v; x, t)dv, / = 0. 1 ^3^
Employing system (2) , we obtain
SDj M du ^ _. dn , au ^ _ (4)
In order to study nonlinear plasma motions, system (4) is more advan- /144
tageous than the initial system of equations (2) , since it includes tenns
which are only dependent on x and t, while equations (2) also include the
electron velocity distributions. In this sense, equations (4) are similar
137
to equations of hydrodynamics, although — in contrast to hydrodynamics
which operate with a finite nimiber of quantities — they Include an
infinite number of "hydrodynamic quantities" n, D- , u.
We are Interested in simple waves, i.e., those plasma motions for
which the perturbations of all quantities characterizing the plasma are
propagated at the same velocity — in other words, for which each of the
functions X [X = u, n, D . , F(v)] satisfies the following equation
Id . ,,, .^ a
|+K(^,0|)x = 0.
In the case of simple waves, as is well known, all of the quantities X
can be represented in the form of a function of one of them (for example,
n) , which in its turn is a function of x, t. System (4) thus changes into
a system of customary differential equations for the functions Dj (n) , u(n),
and the phase velocity V(n) is determined from the solvability condition
of this system. After simple transformations, we obtain
du ^ !j . '^^ -^ .
dn ^ n ' dn ^ Di ' (5)
where e = +1 (e = -1) , if the wave is propagated in the positive (negative)
direction of the x axis.
The determination of the electron distribution function in the case of
a simple wave may also be reduced to solving the customary differential
equation. Rewriting the kinetic equation (2) in the following form
dF{v) M _ (V ~uY dF{y) _
dn ' m n ^x^^x
and introducing the notation
F[vl; vr, n)=BF{v; n)
[v^. = (Vy, Vg.)], we obtain
where the function 3(n) satisfies equation /145
dp , 2IVI V\ f.
- — . — = u.
an. m n
138
(7)
We should point out that Landau damping of sound waves was not
taken into account when the initial equations (2) were derived. There-
fore, it is necessary that the wave amplitude An not be too small,
An/n >> (m/M) /2, in order that equations (2), and consequently relation-
ships (4) - (7) may be valid. The role of nonlinear phenomena in the
development of the perturbation is much greater in fulfilling this condi-
tion, than is the role of sound damping.
Development of a Perturba tion Having a Finite Amplitude
The system of equations for "hydrodynamic" quantities (5) , together
with relationships (6) and (7) , enables us to study the direction of the
change in quantities characterizing the plasma (including the electron
distribution function) and to trace the development of a perturbation
having a finite amplitude.
First of all, let us determine the manner in which the electron dis-
tribution function changes in a simple wave. It follows from (7) that 3
decreases in a contraction wave, and increases in a rarefaction wave.
Therefore, for values of v for which d F/v^ v^ < 0, the number of electrons
having a velocity in the (v, v + dv) range increases in the contraction
wave, and decreases in the rarefaction wave. Conversely, at values of v
for which3F/Vj^v > 0, the number of electrons with velocities in the
(v, v + dv) range increases in a rarefaction wave, and decreases in a con-
traction wave. In particular, if the initial electron velocity distribu-
tion has a spike encompassing a small velocity region, along with a maximum
for Vjj = 0, the spike shifts to the region of larger (smaller) values of
|vjj| as the contraction wave (rarefaction) moves.
Let us dwell in somewhat greater detail on the case of Maxwell distri-
bution. Employing system (5), we may state that in this case Vg, D./n do
not depend on n, and consequently are motion integrals. The electron tem-
perature and the distribution function F/n, which is normalized to one
particle, do not change during wave propagation. Thus, in the case of
a Maxwell velocity distribution of electrons it is valid to describe a
two-temperature plasma by means of isothermic hydrodynamics .
In order to determine the manner in which the form of the sound wave /146
changes, it is necessary to compute the derivative dV/dn [see (Ref. 7)].
Employing the system of equations (5), and asstmiing, for purposes of
definition, that e = 1, we obtain
139
Depending on the electron distribution function, dV/dn may be positive,
negative, equal to zero, or alternating [positive for single values of
the parameter g and negative for other values of this parameter, see
formula (6) ] . (We should point out that the derivative dV/dn is always
positive both in customary and in magnetic hydrodynamics.)
If dV/dn > for all values of the parameter g, then (just as in
customary hydrodynamics) points with a large density move at a large velo-
city. Therefore, discontinuities arise at the contraction sections.**
Self-similar waves are rarefaction waves. In particular, this possibility
exists for a Maxwell electron velocity distribution and for a distribution
in the form of a step F -v (vQ(n) - v^) , G (x) = -^ (1 + sign x) .
If for all values of g, dV/dn = 0, all the points move at the same
velocity during wave propagation. Therefore, the wave profile is not
deformed and no discontinuities arise. Employing equations (5) and (7),
we may state that the velocity of two-temperature sound and the quantity
B '2 change in an inverse proportion to density, V n = const, Bn^ = const.
The case dV/dn = is realized, in particular, for a Cauchy distribution
F "^ {vo(n) + v2}-2.
If dV/dn < (independently of the value for the parameter g) , then
points with a large density move at a low velocity. Therefore, discon-
tinuities arise in the rarefaction sections. Self-similar waves are con-
tractions waves. This possibility is realized, in particular, if the dis-
tribution function is the superposition of two Cauchy distributions
F- V, («) [v' + v', (n)}-'+ V2 («) [v' + vl {n)]-\
We should note that in this case the velocity V increases in the rarefac- /^^^
tion wave, and decreases in the contraction wave.
Finally, let us discuss the case when dV/dn may be both positive and
negative, depending on the value of the parameter g. For purposes of
definition, we shall assume that dV/dn > in the case of g > gj^ and
dV/dn < in the case of g < gi, where gi is a certain critical value of
the parameter g. When a contraction wave moves in such a plasma, the
** We have employed the term "discontinuity" to designate the narrow
regions in which the gradients of the quantities characterizing the
plasma become so large that the initial equations (2) are not
applicable. In the case of a/X ^1, sound dispersion must be taken
into account. With a further increase in the gradients, multi-flux
flows [see (Ref. 4)] or Shockwaves may arise in these regions.
140
"apex" of the wave (points with the density n > nj , where n^ is deter-
mined from the equation 6(ni) = 3i) lags behind the "base" of the wave
(points with n < nj) . Therefore, the density at the point of the dis-
continuity which develops from such a wave cannot exceed nj. A discon-
tinuity may also arise during the motion of a rarefaction wave. Thus,
the density cannot be less than n^ at the discontinuity point. If, on
the other hand, dV/dn < in the case of B > 62 and dV/dn > in the
case of 3 < 32» then — as may be readily confirmed — the density at
the discontinuity point cannot exceed n2 when a discontinuity develops
from a rarefaction wave, and cannot be less than n2 when a discontinuity
develops from a contraction wave (the critical density n2 is determined
from the equation 3(n2) = 32)'
Both of the above possibilities may be realized, in particular, if
the electron velocity distribution is a superposition of two Maxwell dis-
tributions, "hot" and "cold",
F{v; n) = vi (n) exp { - ^ j + v^ (n) exp { - ^'} . T^ » T,.
For small density values (n<^{aiT'i'/a2T'2'Y'^^'), according to equations (6),
(7), Vi = ai/i; V2 = a2«^"'^' («!. ^2 — constants). It may be readily confirmed
that in this case dV/dn > in the case of n < nj and dV/dn < in the case
of n > ni , where «, = (H^ . ^) . For large density values {if^ a'iT'i'/a'2T'2')
\ °2 t/'J
Vj = airt^'^'^'; V2 = a2« («!. 12 — constants). In this case, dV/dn < for
n < n2 and dV/dn > in the case of n > n2, where n2 = —.{TjTi)'''.
2a 2
Let us investigate the motion of a two-temperature plasma arising
during its uniform contraction or expansion [similarly to the problem of
the plunger in hydrodynamics, see the monograph (Ref. 7)]. We shall
assume that the plasma occupies the half space x > Vot, which is uniformly /148
limited by a moving plane. (Such a boundary may represent, in particular,
the region of a very strong magnetic field.) As is well known, only self-
similar waves (in the absence of shock waves) can be steady motions of
a uniformly contracted (or expanding) medium. If dV/dn > 0, a self-
similar wave (which is in this case a rarefaction wave) arises during
plasma expansion (Vg < 0) . If dV/dn < 0, a self-similar wave (which is
in this case a contraction wave) arises during plasma contraction (Vq > 0) .
Employing formulas (5) , (6) , (7) , we may relate the change in all
the quantities X characterizing the plasma in a self-similar wave with the
"plunger" velocity Vg. Assuming, for purposes of simplicity, that
141
I'n. .r. 0/+I Vo.
s
Vq « V , we obtain
REFERENCES
1. Tonks, L., Langmulr, I. Phys. Rev., 33, 195, 1929.
2. Gordeyev, G. V. Zhurnal Eksperimental'noy i Teoreticheskoy Fiziki,
27, 18, 1954.
3. Akhiyezer, A. I., Lyubarskiy, G. Ya. , Polovin, R. V. Zhurnal
Teoreticheskoy Fiziki, 29, 933, 1959.
4. Bedenov, A. A., Velikhov, Ye. P., Sagdeyev, R. Z. Yadernjry Sintez,
1, 82, 1961.
5. Klimontovich , Yu. L. , Silin, V. P. Zhurnal Eksperimental'noy i
Teoreticheskoy Fiziki, 40, 1213, 1961.
6. Nekrasov, F. M. Zhurnal Teoreticheskoy Fiziki, 32, 663, 1962.
7. Landau, L. D., Lifshits, Ye. M. Mechanics of Solid Media (Mekhanika
sploshnykh sred) . Gosudarstvennoye Izdatel'stvo Tekhnicheskoy i
Teoreticheskoy Llteratury, Moscow, 1953.
142
NONLINEAR PROCESSES IN A UNIFORM AND ONE-COMPONENT PLASMA
N. A. Khizhnyak, A. M. Korsunskly
Nonlinear solutions of a one-dimensional kinetic equation without a
collision term, which depend on x and t by the combination ? = x - Vot,
where — Vq is the constant wave velocity, were compiled in (Ref . 1) /149
and studied for several cases in (Ref. 1-4). The conditions at which
these solutions may be realized were found in (Ref. 4), and a limiting
transition to small oscillations was performed.
General Theory of Nonlinear Waves
This article investigates the more general nonlinear solutions of
a one-dimensional kinetic equation without collisions
dt ^ "■ dx dx du — "'
where <}) is the potential of the self-consistent electric field with the
factor — , determined by the Poisson equation
where ng is the unperturbed density of ions whose mass is assumed to be
infinitely large. It is assumed that the distribution function f depends
only on the variables u and ^. Then the electron density
< n ) =^fdu==F(<f), (3)
the density of the electron fltix
< nu ) = Ju/d« = 4>(9)
(4)
and the energy density
(n^ > =J/(u)f-'d«=^(cp) (5)
are explicit functions of only the potential (j).
We shall show that in this case the main plasma characteristics may
be compiled within the framework of a hydrodynamic approximation, and
that the electron velocity distribution function may be found relatively
simply. From the equation of continuity
143
we find that the potential (}> must satisfy the following equation
|+l^o(T)g = 0. (6)
where Vq = — ^ is a certain velocity of longitudinal waves in the plasma
which depends on the potential (f). It can be readily seen that the specific /150
solutions depending on x - Vot, where Vq = const, are special cases of
equation (6) in the case of Vo(i)>) = const.
From this point on, it will be assumed that the function Vo(<|)) is
given, and it may be employed to formulate the solutions both for the
equations of the hydrodynamic approximation and for the kinetic equation.
The equations of the hydrodynamic approximation
may be reduced to equation (6) and to the equation for hydrodynamic velo-
city v(x, t). In actuality, since
( n >
it follows from expressions (3) and (4) that v = v((t)) . Therefore, the
Navier-Stokes equation (Ref. 7) has the following form
— r°!? -1- ^fl — ^1 I dp d<f
and is identical to equation (6) , if only v(<|>) is determined by the equa-
tion
-^;+itizi.^v^(,). (8)
¥
The electron plasma density <n> may be found from the specific wave
velocity VgCtf') and the hydrodynamic velocity v((j)) from equation (8), by
means of the following relationship
-f "^ df
„ , . J a (9) - I'. (V)
144
which follows from the equation of continuity. Thus, the dependence
of the main hydrodjmamic plasma parameters on the potential (|i is always
formulated in quadratures for a certain Vo(i))).
The potential of the self -consistent field (^ is determined according
to equations (6) and (2) . The general solution is found in the following
form from equation (6) according to a certain function Vo('l')
? = ? (c).
where c = c(x, t) is the equation of its characteristic. The characteris-
tics of a quasilinear equation in partial derivatives (6) may be com-
piled according to the well known method (Ref. 5).
We shall regard t, x and (j) as functions of a certain parameter s. /151
Then the parametric equation of characteristics can be written as
follows
dt , dx ,/ .V. d<f
from which it follows that <\> equals ^q and does not depend on s. There-
fore, X = Vo(<(>)s + XQand t = s + tg, where Xq, tg and (J>o are the values of
t, X and (f) on a certain line s = through which the characteristics pass.
In particular, if primary Interest ia directed toward the development of
moving waves, which is caused by the nonllnearity of the medium, xq, to
and ^Q must characterize a given Initial moving wave.
Let us assume that at the initial stage of the process there is a
given moving wave with a constant phase velocity wq . We then have
where ^q^i) characterizes the initial dependence of the field potential
on time. In this case, we have
/ = s + t; ^ = ^0(^0 (^)) « + K'o'^; <P = To i^)
and after excluding s we find t from the following equation
which determines the characteristics of the quasilinear equation (6)
T = t(x, t) . Using the characteristics from equation (2), let us deter-
mine the field potential in the plasma <()(x, t) and all of the hydrodynamlc
parameters of the medium.
145
\%
We can find the electron velocity distribution function from the
kinetic equation (1). Actually, since equation (1) is transformed into
the following form, together with equation (6) ,
[«-l^o(?)]|-| = 0.
the equation of characteristics
enables us to find the particle velocity distribution function in a
general form from the potential (j) and the kinetic velocity u. Since the
potential ^ may be compiled within the framework of the simpler hydrody-
namic approximation, the specific dependence of the distribution function
on the coordinates and time has been established.
Propagation of Waves in Media_wit h Li near Dispersion /152
We shall assume that the given law for the dependence Vq ((j)) of the
nonlinear wave phase velocity on potential determines the law of the
medium dispersion. We shall study nonlinear, nonstationary waves in a
medium with linear dispersion
where Vq and y are certain constant parameters.
We shall disregard the plasma pressure, so that equation (8) assumes
the following form
■i = -^ + ^o + -(?-
The dependence of the hydrodynamic velocity v of the plasma on the poten-
tial can be determined according to the following relationship
where X = v - Vq ; c — integration constant. In the case of v = Vq, dis-
persion disappears, i.e., in the sense that Vq, is the limiting plasma
velocity at which the electric field potential vanishes. We then have
?=^[(T5C+l)-eT']. (9)
146
Since
dv 1
the electron plasma density may be found according to formula (9)
» df
J [U— V, (f)J' J [x— 1«>1" "A.
n = «oef«. = n„e'
Since
J [X — 7?]^ J 1— e'" 1 —
F(<p)=rto
1— fiT'
where xo is the value v = Vg at which the electron density equals the ion
density.
Consequently, the Poisson equation can be written as follows /153
d^<P 2 1
and X may be found from relationship (9) .
Let us determine the characteristic of equation (6) for the linear
dispersion law
Let us assume that at an initial moment of time the field is switched on
whose potential increases linearly with time:
9o {'^) = To -^^
We then have , ~, \ t
from which it follows that
147
Consequently, the characteristic Is real only up to a certain value x^^gy^^^j
which depends on time
bound ^^ L "PoTf J
If ^>Xt^ound> *'^® signal which is switched on in the case of t = has still
not reached the point under consideration. If x<X|jq^j^(J, the characteristic
is determined in more than one way (it has two values) . The requisite
value of the characteristic is determined by additional considerations.
The propagation rate of the front boundary is
Vbound=lP°""'=^^4i^+2M^.
2 ~ AT
Consequently, for y > the front velocity increases with the time, while
in the case of y < it decreases.
The velocity of a point with a constant given potential, as may be
seen from (9), is determined by the wave phase velocity Vo((t)), i.e., by
the specific value of the potential ^. We may formulate the potential and
the electron velocity distribution function by the hydrodynamic parameters
which are found.
Electron Velocity Distribution Fimction in the /154
Case of a Square Dispersion Law
Let us find the electron velocity distribution function in the case of
media with a square dispersion law
The equation of characteristics
dip
dX
g = -x,±i/=:2?.
148
where xi = u - Vq; u — electron kinetic velocity in the plasma, can be
rated by substit
We finally find
integrated by substituting (j) = x?t
u-\\,r \ -2y A -2y J
^ 1/ ("-I'o
X
21/' -2?
/ 5 — 1
for a positive sign of the root, and
--. I
1 + 1^5
(10)
= Ci == const
(u-V,)^\ _2y ^ /" -2y ]
2*r L(u-Ko)^ y («-l'o)' 'J
21/11=^
^ _ y (u- v^r
X
-, 1
K5+1
21/ -^y
(11)
= Co = const
for a negative sign of the root.
The solutions of (10) and (11) enable us to compile the distribution /155
functions in the case of i}" "*■ which change into a Maxwell distribution.
On the other hand, all the distributions changing into Maxwell distributions
in the case of (J) ^0, in regions with a non-zero electric field, have the
following form
/ = y4 exp Ci for u > Vq,
f = A exp C2 for u < V ^.
Consequently, electrons have velocities lying outside of the boundaries
«>Vo +
2V-2<t
/5— 1
and
7^5+1
149
The electrons whose velocities are included within
" /5+1 ° /5— 1
are damped by a wave, and are not included in the distribution functions
of (10) and (11) . The relationships obtained enable us to study the
collisionless transition of electrons from the region of trapped particles
into other plasma electrons and the associated energy distribution.
REFERENCES
1. Akhiyezer, A. I., Lyubarskiy, G. Ya. , Faynberg, Ya. B, In the Book:
Scientific Reports from Khar'kov State University (Uchenjrye Zapiski
KhGU) , 6. Izdatel'stvo Khar'kovsk. Universiteta, Khar'kov, 73,
1955.
2. Faynberg, Ya. B., Nekrasov, F. M. , Kurilko, V. I. In the Book: Plasma
Physics and Problems of Controlled Thermonuclear Synthesis (Fizika
plazmy i problemy upravlyayemogo termoyadernogo sinteza) . Izdatel'stvo
AN USSR, Kiev, 27, 1962.
3. Nekrasov, F. M. Zhurnal Teoreticheskoy Fizlki, 33, 7, 1963.
4. Khizhnyak, N. A. In the Book: Plasma Physics and Problems of Con-
trolled Thermonuclear Synthesis (Fizika plazmy i problemy upravlyaye-
mogo termoyadernogo sintez^, 1. Izdatel'stvo AN USSR, Kiev, 31, 1962.
5. Smirnov, V. I. Course on Higher Mathematics (Kurs vysshey matematiki) .
Vol. 4, Fizmatgiz, Moscow, 1956.
150
INDUCED SCATTERING OF LANGMUIR OSCILLATIONS IN A PLASMA
LOCATED IN A STRONG MAGNETIC FIELD
V. D. Shapiro, V. I. Shevchenko
This article investigates the nonlinear interaction of harmonics in /156
the long wave spectral region of Langmuir oscillations (kvDe ^"^ ^^ • '^^
linear damping of these oscillations, which is caused by the interaction
with resonance particles, is negligibly small. It is assumed that the
plasma is located in a rather strong magnetic field, so that the plasma
particle oscillations are possible only in the direction of the magnetic
field which is parallel to oz.
The initial system of equations for the distribution functions of
electrons and ions and the electric field has the following form
5/1 ,,a V Sfl
(1)
X exp [—ilw^ ^ + u)_ — mj\ t);
\ k-q q k )
k a J k \^J
T i:k ^ -^ k
k k k
(fg is the background distribution function whose change with time can be
disregarded, due to the small nimiber of resonance particles ( — -—j^Xd^ <^ l)
and the small dissipation of oscillation energy during scattering). The
notation in equations (1) - (3) is standard; summation in equation (2) is
performed for plasma ions and electrons. We obtained the following rela-
tionship for damping frequency and decrement, disregarding the nonlinear
terms in the first equation, from equations (1) and (2):
o_. = <o„,(l + I ^=x^^] cos e; T_. = I- . ^ cos 5^^
(4)
CO— •■
k
(e is the angle between the direction of oscillation propagation and the /157
magnetic field) . The nonlinear interaction of harmonics leads to a change
in the oscillation spectrum due to processes of wave decay and scattering
by plasma particles. The laws of conservation must be fulfilled in the
c. c. = complex conjugate.
151
case of two-plasma decays
(u_» + (o_ = 0)^; ki+k2= k. (5)
->• ->-
Assiomlng that kj and k2 lie in one plane, we obtain the following condi-
tion for the spectrum (4) from (5) :
k-i cos G] it ^2 COS Bj
Vk^i -f A| ± 2A,&2 cos (Gi — 02)
= COS 9i + cos 62. (6)
Decay is possible if k^^ ^ 0» ^22 < 0, which corresponds to the sign " — "
in condition (6). In several cases, the spectrimi of Langmuir oscillations
in a strong magnetic field is a nondecay spectrum, particularly if it is
close to a one-dimensional spectrum O^ 2l 62*.
Nonlinear wave scattering is caused by the interaction of plasma parti-
cles with the beats of different frequency. This process becomes signifi-
cant if the condition *'' '' 'f^ d„ is fulfilled. The beats which are
the cause of wave scattering cannot arise due to decay, since the following
condition is fulfilled for the waves formed during the decay
(J) — (i)_^ <^_^
k[ ft. ft _ "0 -^ „
Therefore, in this case the processes of wave decay and scattering are in-
dependent. The transformation of the oscillation spectrum due to the non-
linear scattering process will be subsequently investigated.
In solving the nonlinear equation (1) , we shall employ the method of /158
perturbations. Substituting f", which is found from the linear theory,
k
in the nonlinear terms of this equation, we may employ (2) to obtain the
following formulas for the electric field amplitude and the distribution
function in the second approximation
Ion oscillation branches are not examined. Actually, in the approxima-
tion under consideration, decay of a Langmuir wave into a Langmuir
ion-sound wave is possible. However, the energy primarily remains in
high-frequency oscillations (Ref. 1).
152
X
ft— 17 9 \ gy
dv;
f=-S
k-9
1
X
I ft-? 1 I
X
^4^ r i_
ft—? fl \ 9' -I
dy.
(7)
In the third approximation, the system of equations for determining
the electric field amplitude and the distribution function has the
following form
1-
dt
+ i {kzVz — U>-
) r + -^£_, eft;, = _ _« ^ £11) ^
" k ' l^a kz '"a - \ ft-?.
+
a/11'
u)_. — a)_
q q kj
(8)
f2) (2)
where E'-.,. , f ^ are determined from formulas (7). We obtain the kinetic /159
k k
equation for waves from these equations by simple computations:
-^ = i^E'^ - / 2 // (k, 9. x\ E E X
"' k kz -— \. I k-qz q — xz
q,x
X eXp f i /"ffl-, _ + 0)_^ ^ + 0)^ iO^\ t\.
(9)
153
where y"^ is the decrement of Landau linear damping:
k
// {k, q, x)
k—q q — X X fe
\ ft— 9 q—7. X J
- X
X (
X
k — q q — X X
d I 3d,
?^^z
^^z \ '-z^z — «■-. ,
X
17 — X X
.3
dy +
---rV
\ <7— X X j
;9^^'"'
4< p 1
J *— ? q—
q — X X
dv.
I 1 \ dfl
Qz"z —"> ► — "-. (*z — 9z) "j — »-. ^1 3^2
?— X X k~q I
X
X
4iteS
d I 3"'
a '"p
m^ J 92^2 — "^ - — '
dv^ \ x^u^ — '"-
\dv )
9— X
Multiplying (9) by E , averaging over time (only the terms with
kz
-^ ->■
/160
X = k and x = q - k, which change slowly with time, remain in the right-
hand side) and combining the equation obtained with the complex conjugate
one, we can write
dt
kz\ k\ kz] -^k q \ kz\ \ q \
(11)
.Y^ = ^Im ( H{k,k~q,q)+H{k.k—q,k) )
1 V 4tei
dfo
dv.
k
('■'■-i)'
X
X
^^ + fc^x
{,z^z-^^)[{K-^z)^z-^^+<^^\ \X_-;\
154
X ^^rx
V ft 9 )
dv] ). (12)
In computing the Integrals Included in equation (12) , we assume that
the following conditions are fulfilled: o) >> kv^, which corresponds to
k
weak wave absorption, and f^^— w » which is necessary in order that a sig-
k q
nificant number of plasma particles may participate in the interaction
with beats of different frequency. Let us investigate the case when the
contribution made by Ions to the plasma polarizability at a different
frequency is negligibly small — i.e., the condition , __ r^- 'v
'^ m bn* C 1- is fulfilled. Then, assuming that the distribution func- /^^^
"^i De
tion of the plasma particles fg is a Maxwell distribution, we obtain the
following from equation (12) after very cumbersome computations
T" (k, 6. <7. e') = - ? l/l • ;;4-cos 6F,^ f^>^.. X
/, , 2 ( cos e - COS 9' ) \ ( m <4jcos^j-cos£)^|
^ r + ¥ ■ (k-^- q^) l-'^, cos 9] ^^P l~ 2T, ' (k - qr cos^ 6 f
The increment y \k,e,q,0') which determines the Induced scattering rate
of plasma oscillations in k-space differs considerably from zero only in
the narrow range of angles: |8' — 6|:S — in the case of 6 t= and
1 9 _ 6' I < t/ ^ in the case of 9 :^ .
Thus, during scattering the spectrum which is initially one-dimensional
remains close to a one-dimensional spectrvim. In the case of 9' - *-*-, it
coincides with the increment obtained previously in a one-dimensional model,
within an accuracy of the factor cos (Ref. 2, 3). It increases somewhat
with an increase in 9 - 9' ^*'^~ up to f'^a^^yi' W+W^X^^ '^^'^' ^^^
then rapidly decreases to 0.
155
illllllil
In the real case of a confined plasma the spectrum of the values
Vrj.
is always discreet, and for a sufficiently small ratio — =^, only one
v$
possible value 6' = 9 can enter the angular range 0' - in which y
is large. The scattering of plasma oscillations in a strong magnetic
field then takes place, in fact, in the same way as in a one-dimensional
model, in contrast to scattering of oscillations in a plasma which is not
located in a magnetic field. In the latter case, scattering at large
angles is possible, and, if the angle between the wave vectors of two
interacting waves considerably exceeds , y^^i^icreases by a factor of
1 Q Q
as compared with y in a one-dimensional model (Ref. 4, 5). The
k2x2
De
contribution made by ions to the plasma dielectric constant is quite sig-
("0/ cos'' 9
nificant when the condition ~, n- ~:^ 1 is fulfilled. We should note
I ^ 7)
that this condition is fulfilled most readily in the case of 2i S, when
the difference to - o) is at a minimum: m., — u)_ =^ -h- u'o^ cos ()fe* — 9^)^ J ^^'^ /162
k q * , ^
wq,- cos' 9 m, 1 /«,, 1 m,
K~^^«irf ^ «('fe^* Assuming that the conditions TR}' ^>^ ^ m^^
X . <g 1, are fulfilled, we obtain the following expression from (12) in
the case of 6 = 0' for y^^
T"(/^.0.,.6) = -^y ^. ^^-j ,-^_^. _-_^_. (14)
i.e., in this case y increases somewhat as compared with (13). In the
2
9' /cos G _ cos 8' ~ :il\ ^ -'' -
m.
case of G =5t 0' /cos — cos 8' —\ ^ x 1 and when the condi-
tion —
«<. 1
(13).
■C 1 is fulfilled, y is determined in this region by formula
We should recall that this investigation pertains to a case when the
plasma is located in a magnetic field which is so strong that the plasma
particle oscillations are only possible in the direction of the magnetic
field. The condition of "magnetization" of the electron component, as is
customary, has the form coog << Wjje* i.e., it is fulfilled for field
strengths which are not too large. In this case, when the ions make a
156
significant contribution to the dielectric constant of the plasma, it
must be required that the ion plasma component be "magnetized". The
corresponding condition is harder:
— <0_» I < U)//,a . e ; <"//« > Woe — ' ^^^oe-
q I '"«
Let us turn to certain general characteristics of the nonlinear
change in the wave spectrum. Since the Increment of nonlinear scattering
7^ (k,e,q,e') in the one-dimensional case 6=0' changes sign when k^ q
is substituted [see formulas (13) , (14) ] , the change in the total oscilla-
tion energy during scattering in the approximation under consideration
equals zero
fc k'q ^ -^'
Employing formulas (13), (lA) , we may also readily see that the non- /163
linear interaction of harmonics leads to a transfer of energy along the
spectrum to smaller k:
S^l^-r^ Y.\^^-9UHk,^,,,n\E^\Y-;\'
(16)
* k q
The total oscillation energy during scattering by plasma particles
changes in the subsequent series with respect to k^X'^ , since in this
De
approximation an addition to y^ appears , which is symmetrical with respect
to the k 4?" q substitution. Assuming, for purposes of simplicity, that
the oscillation spectrum is one-dimensional, we obtain the equation for
the change in the total energy in the spectrum:
avlr:.!*. V fn T. "''^'D' "^
1 3
k-\%.
k
(17)
The second term in this equation which describes the oscillation energy
change during scattering becomes more significant than the first term,
which characterizes the oscillation energy change as a result of their
interaction with resonance particles, when the following condition is
fulfilled
157
3 '
1 — 2 « 2*^X2
oO ^,0^10
(18)
i.e., for relatively small oscillation amplitudes |e, p « NqT, if the
parameter is fairly small.
The possible dissipation of oscillation energy when they undergo
nonlinear scattering by plasma particles was pointed out in (Ref. 6, 7).
However, for Langmuir oscillations this phenomenon is k^X^ times less
than the change in the field intensity in separate harmonics in the spec-
trimi due to energy transfer.
REFERENCES /164
1. Orayevskiy, V. N., Sagdeyev, R. Z. Zhurnal Teoretlcheskoy Fiziki,
32, 1291, 1963.
2. Drummond, W. E. , Pines, D. Yademyy Sintez. Appendix, 3, 1049, 1962.
3. Shapiro, V. D. Authors 's Abstract of Candidate's Dissertation
(Avtoreferat Kand. Diss.). Ob ' yedinennjry Institut Yademykh Issle-
dovaniy, 1963; Zhurnal Eksperimental'noy i Teoretlcheskoy Fiziki,
44, 613, 1963.
4. Gorbunov, L. M. , Silin, V. P. Zhurnal Eksperimental'noy i Teoretl-
cheskoy Fiziki, 47, 200, 1964.
5. Gaylitis, A. K. , Tsytovich, V. N. Zhurnal Eksperimental'noy i
Teoretlcheskoy Fiziki, 47, 1468, 1964.
6. Kadomtsev, B. B., Petviashvili, V. I. Zhurnal Eksperimental'noy i
Teoretlcheskoy Fiziki, 43, 2234, 1962.
7. Karpman, V. I. Doklady Akademii Nauk SSSR, 152, 587, 1963.
158
NONLINEAR THEORY OF LOW FREQUENCY OSCILLATIONS EXCITED
BY AN ION BUNDLE IN A PLASMA
D. G. Lominadze, V. I. Shevchenko
As is well known, in the case of the interaction between a bundle
of rapid electrons and a plasma, the energy lost by the btmdle during
relaxation changes into thermal energy of electrons in the plasma and in
the bundle, and into energy of high frequency Langmuir oscillations
(Ref. 1). When investigating the possibility of heating the plasma ion
component during bunched instabilities, it is of interest to investigate
the excitation of low frequency oscillations by the bundle, in the non-
linear approximation. A previous article by D. G. Lominadze (in collabora-
tion with K. N. Stepanov) investigated the linear theory of low frequency
oscillation excitation in a plasma located in a magnetic field by an ion
bundle. In a strongly non-isothermic plasma (T^ >> T^^) , during the passage
of an ion bundle, longitudinal longwave/ 4- J_a << l\ oscillations may be
excited, whose frequency is determined by the following relationship
'l = ^ (">' + <d ± y [("^s + '^hy-^«i ^°^' H '''' (1)
U)
Wl^e^e »?. Un„e- . ^2 _ T.
1 +
M ' De \T^nJ^ '
Wjj^ g is the Larmor frequency of ions and electrons, respectively; — /165
the angle between Hg and the direction of oscillation propagation. If
T^ "^ Tg, the ion bundle can excite longitudinal shortwave oscillations
f-e^^)
in the plasma with the frequencies (Ref. 2)
U)„ = nu)w,(l + tpn); '^n =
(-9
tt = 1, 2, . .
e h -^rz^ (2)
k|yxi
where y^^ = ; Ijj(yj) is the Bessel function of the imaginary argument.
0)2
Hi
The process by which bunched instabilities develop may be
159
dynamic) stage, the bundle remains monoenergetic ( « 1, v,
k
qualitatively divided into two stages (Ref. 1). In the initial (hydro-
Tl ~
thermal velocity of bundle ionsl and instability develops veiry rapidly;
j^j '^(/Iq. "i — densities of plasma and bundle, nx « ng, co —
excitable frequency). For rather large oscillation amplitudes, the
thermal energy in the bundle is so large I rr~ — 1 1, that the bundle
relaxation may be investigated in the quasilinear approximation (quasi-
linear stage) . The time required for the development of this instability
T^O 1
TniiaGT is on the order of — • — .
quasx j^^ (jj
This article investigates the development of instability at the first
stage, the most unstable oscillation branches are found which produce the
dynamics of the instability development, and the change in the macroscopic
parameters of the bundle and the plasma is determined (thermal energy, di-
rected velocity) in the case of instability. In a strong magnetic field
("^i ^^ '^s^ it is possible to trace the development of instability at the
quasilinear stage and to determine the state at which the bundle and the
plasma arrive as a result of the quasilinear relaxation process.
Excitation of Longwave, Low Frequency Oscillations /166
(Hydrodynamic Stage)
We shall assume that at this stage the following condition is ful-
filled
-e«^'-t-«' (3)
k k
(6u is the change in the bundle velocity at the initial stage) , at which
the dispersion equation of longwave oscillations has the form
"01
2-sin2 9=0.
When investigating the dispersion equation (4), we can examine two cases.
160
1. a)jj£ « 0) (weak magnetic field). During Cherenkov excitation
(o) = k^ Uq), oscillations with the frequency
(,+l.|3i„.»).
have the largest increasing amplitude increment. The maximum value of
the corresponding increment is
l^(^V\>.rn,V.9. (5)
The formation of instabilities in this case is possible in the case of
cos 6 < — ^, if Un > v„ , and for any cos 6 , if u„ < v„ I v„ =^/_§. 1.
Up OS ° ^ V Tm /
2. a)jj£ >> (iOg (strong magnetic field). In this case, ion-sound waves
the frequency to = w cc
increment of these waves is
with the frequency to = w cos 6 have the largest increment. The increasing
V
(6)
Excitation of oscillations is possible if — H. > 1.
"0
Relationships (4) - (6) were obtained by disregarding damping by /167
plasma electrons, which is valid if the following conditions are fulfilled
If condition (3) is fulfilled, all of the bundle particles are in resonance
with the wave, and its hydrodynamic description is possible by means of
the moments of the velocity distribution function
^ = 1^ v^dv; n=i-J {^' - "^) {"" - "*) ^S^''-
We can obtain the equations describing the change in these quantities with
time from the kinetic equation of the Fokker Planck type, which is derived
in (Ref. 3)
at -dviY-ii'dv^)' (7)
161
3 = e, i pertain, respectively, to plasma electrons and ions and bundle
ions; a^j^ — diffusion coefficients in velocity space:
e^ V r p 12 *^^ V" "°'^" ^^'^^ rr 4- T V
n=l
(8)
n=l
'_„ + r„].
Here we have
"±n — TT
/(o':^-;i^u^± «<»„,.y^
The frequency (ji)-> and the increment Y"^ ^'^^ determined by the disper- /168
K k
sion relationship of the linear theory (4) .
We obtain the system of equations for the change in the "bundle"
parameters due to the development of instability from formula (7)
^ _ 2^2 y If IM ^ ^
dt AT- ^ I k\ \ k^
('-"•- "r)
[(v.-.y'H-e]
~ fc2 /I... -^
^^ k'^ 4a.
«i
X
7-*
k
7-.
*
L^ ft ) k \ k Ik.
dt
* ^ *'' ft
X
+ ■
(*."o - -1 - »«,)'+ 7l (*,«o - <»: - ««,)'+ d
(9)
162
The change with time of |e | ^ is described by the following equation
I * „. ,^ ,2 (10)
(it
k I k
We may obtain the equation for the change in the plasma ion para-
meters from formulas (9), assuming that u^ = 0. The condition for the
kAv^
applicability of a hydrodynamic description of plasma ions — << 1 is
less hard in the case of n^ « ng than equation (3) , and this description
is applicable throughout the entire development of instability.
Integrating equations (9) with respect to t, we obtain the equations
determining the change in the energy of directed bundle motion and in the
thermal energy of ions of the bundle and the plasma.
In the case of a weak magnetic field (wj,. << Wg) , we have /169
ni/wwoSw = - 4^ • ^Sp^l';
k ft
In the opposite case (Wrj. >> Wg) , we have
o^„,W \ t. ^^7«K^ . ft. "C>^-,, A, (-Li)
„2
'-7 •
163
The change in the electron distribution ftmction in the case of
instability is determined according to the following equation
<}fo_ dV 1 r^tiP [2:^.__^iL 1°
4^J^*l^rr*^"(<oj-*Af+7i
dt - dvAm'- • (27:)' I "" r r "' fcoi _ A o \2 + ^l dv
(13)
The phase velocities of the excited oscillations change between Vrj.^ <<
<< Vj, << Vrp . Therefore, the main portion of plasma electrons is in /170
resonance with the waves v >> v^. The change in the electron distribu-
tion function for these v is described by the equation
[ k }
Thus, for a change in the electron thermal energy in the case of insta-
bility, we obtain the following in the case of a weak magnetic field
and in the case of a strong magnetic field
«s
I.„.,7-„=_^.l2|£.p. (16)
1- 2
For plasma electrons which are in resonance with excited oscillations,
whose velocities lie in the narrow range — ^ «/^ , we obtain the following
■^Te » "
expression from equation (13)
|-.iwf''»rrr5M"r-*-"-)'^"|- <!'>
Changing to the variable $ =-^( I / ~<i<l] in formula (17) and
representing fp in the following form
164
f^='U-T^),.(^A (18)
where (p (?, ^) 1,^0 = 2"' ft\v±] ^^ ^^^ portion of the electron distribution
function depending on vj_, we obtain the following equation for the change
in (t)(C, t)
dt In. ' m^ £,5
{lVrrS'"T.s}- (19)
The Interaction of resonance electrons with oscillations leads to /171
the occurrence of a plateau in the electron distribution function in the
region of excited oscillation phase velocities. The time required to
establish the plateau may be determined from formula (19) :
*
(A? — the dimensionless width of the plateau).
The change in the energy of resonance electrons is
Mj \/n / ft ft
(21)
i.e., it is considerably less than the change in the total energy of plasma
electrons.
We may determine the balance of energy during the development of
instability from formulas (11) and (15) , (12) and (16) : The energy of
directed motion, which is lost by a bundle, changes into the thermal
energy of particles in the bundle and the plasma and into energy of
electrostatic oscillations. Since conditions (3) must be fulfilled in
the initial (hydrodynamic) stage, we may readily estimate the maximum
energy of oscillations excited at this stage:
r
At this oscillation energy, the time required to establish a plateau in
the electron distribution function, as follows from formula (20), is
165
(23)
i.e., it is a little less than the time required for the development of in-
stability in the initial, quasilinear stages.
Thus, the plasma electron distribution function changes most rapidly
in the resonance region: A plateau appears in this region in the velocity
range in which oscillations are excited at a given moment of time. A
change in the spectral density of oscillations and ion parameters occurs
much more slowly. Under these conditions, the electrons have no signifi-
cant influence on the dynamics of the instability development.
Excitation of Low Frequency Oscillations^
(Quasilinear Stage)
/172
Further development of instability leads to a still greater increase
in the thermal scatter in the bundle, and its distribution function be-
comes so diffused that the quasilinear approximation is applicable.
In the case of to >> ojg^, it follows from the expressions for the
diffusion coefficients (8) that aj_j_ '\' otzz* ^^ ^^"^ '^ '^' '^^ problem is
thus essentially a three-dimensional problem. Let us investigate the
opposite case, a)jj- » tOg, since in this case a_Lj_ << a^z at the quasi-
linear stage, i.e., only longitudinal diffusion is significant.
The initial system of equations for the quasilinear approximation
has the following form
dt " Ov
^'?Krr>(";-
I k I
dt
-*.".) g
r «! '"sdg
*
(24)
(25)
Here g(t, v) is the distribution function of bundle ions integrated with
respect to vj_.
Substituting r— °- Ie p from equation (25) in formula (24), changing
k
166
from summation over k to Integration, and integrating with respect to k
due to the 6-f unction, we obtain
"01
1a1\
dt
(26)
Integrating with respect to v and t, we may write the following
equation
71/2
I sin 0d6 f I £_, (/) p _ i £_^ (Q) pi = 2-k
11*1 \ k I /
X U^ 1 —
_9^2^I. "1 J_x
^ n^(^ j'')-^(o, y')]d^'.
2\Vl O
(27)
where vj is the lower boundary of the instability region caused by damping /173
by plasma ions. We may employ this equation to determine the oscillation
energy at the quasillnear stage:
W =
i^|£.(oo)p=i4^„,j;|^i_ljji^
(oo, V')-
(28)
-g(0, v')]dv'dv.
where g(«>, v) is the distribution function at the end of the quasilinear
stage — the plateau, whose height is determined from the following condi-
tion
^(oo, v)ivo — Vi) = ni; g{co,v)
(29)
v^ and V2 (the upper boundary of the instability region) are determined by
the following relationships
^(oo, fi) = g-,- (yj); g(co, v.)=go(v2y,
(30)
go(v) — the distribution function of the ion bundle at the beginning of
the quasilinear stage; g.(v) — the distribution function of plasma ions
integrated with respect to vj_. We obtain the following from (30)
/2T-
/Jn"n
2r.T,\'/'
-m
««o; t'.-«o(i-(^J'). (31)
Performing integration with respect to v in (28) , we may write
167
The change in the energy of the plasma ion thermal motion may be deter-
mined from (11) :
i^o^Tu = -hl^\E^l4-^=i^''^"l- (33)
I r
The increase in the energy of the plasma electron thermal motion is
k
The finite bundle velocity established at the end of the quasilinear stage /174
is
■ Vt
""= J Vg(ca, v)dv^-^^^^^.
Vl
i.e., the energy loss of the ordered bundle motion
8e=-4rtiM«2. (35)
O
_3
8
The thermal bundle energy acquired during the development of instability
is
n,hT,
- = J^(~. v)[v l^j _d„^__. (3gj
We may employ formulas (32) - (36) to verify the fact that the law of con-
servation of energy is fulfilled
k
Thus, in contrast to the excitation of high frequency oscillations,
the excitation of low frequency oscillations leads to the transfer of a
(4)o.
considerable portion I'^r-jof the bundle energy to plasma ions, and also
leads to significant heating of the ion component.
The quasilinear theory disregards the nonlinear phenomena of oscilla-
tion scattering by plasma particles. As was shown in (Ref. 1), in the case
168
of the excitation of Langmuir oscillations by an electron bundle, non-
linear interaction of harmonics is usually insignificant in the quasi-
linear stage. However, there is a parameter region in which these phenomena
determined the dynamics of the instability development (Ref. 4).
We shall continue to study the role of the nonlinear interaction of
harmonics in the case (which we are considering) of excitation of ion-
sound oscillations by an ion bundle.
Excitation of Shortwave, Low Frequency Oscillations /175
(Hydrodynamic Stage)
When allowance is made for the finite, Larmor radius of plasma ions,
excitation of longitudinal oscillations by the harmonics '^^<*Ht. is possible.
One important feature of these oscillations is excitation in the case of
T^ '^ Tg and propagation almost perpendicularly to the magnetic field.
Therefore, it is more likely that low frequency oscillations are the
reason for anomalous plasma diffusion perpendicularly to the magnetic
field, rather than ion-sound oscillations, for which — << 1.
K
Let us investigate the manner in which the ion bundle and the plasma
parameters change at the initial excitation stage of low frequency oscilla-
tions.
The dispersion equation for shortwave oscillations for conditions (3)
has the following form
Ti V ""'"(C) ""oi 2fi "01 y
n= — oo
X Sin^ 6 == 0.
Waves with frequencies of Wjj = nojg^ (1 + ^^ have the largest increment.
Their increasing increment in the case of n = 1 is
where
- 2 \n,
a{k)
/l(^)'
169
The change in the ordered velocity and thermal energy of bundle ions
may be described by means of (9) , where w and y 3.re determined from
k k
formulas (38) - (39) . Making use of the fact that y^ has a maximimi with
respect to k, we may obtain the following in the case of instability at
the frequency co = cojjj (1 + ^i)
M / *=*,
(40)
' ' ' <"?/.■ L. I'- 1"» ^^«oV ."^
^2 -V 0/ a /.(ft)
J ft=ft.
(kg — the value of k at which y has a maximtrai) • Let us employ the /176
k -^
approximate formulas given in (Ref. 2), and we shall assimie that a(k)
reaches a maximum in the case of -^° ^^ ^i^ } 5;
a(I)^_^^«l; iil^£L<. (41)
Assuming that << )|)i « 1, we obtain the following expression for the
plasma ion diffusion coefficients in the case of instability at the fre-
quency (1) = a)jj^ (1 + ii)i)
a -^. -l-{dk\F |2^^ " ' """'
e-
n=I
7-.
X - ^—
(42)
' + ll
170
II I II
n=l
X_^
k
X
/?(xi)
2
,. X
We obtain the following by means of (42) from formula (7) : /177.
k- d^i^Yrf
(43)
M
We may determine the change in the plasma electron energy by formula (14)
1 „^8r . . = /'4;^ 8^ • (ripj ^^1 ^r r * ^''''^
k" — /- -
Comparing expressions (40) , (43) and (44) , we can see that the
energy of the bundle ordered motion changes primarily into energy of the
transverse thermal motion of plasma ions, and may lead to significant
ion diffusion perpendicularly to the magnetic field. Relationships (40),
(43) and (44) are valid as long as the conditions of a monoenergetic bundle
are fulfilled (3).
The maximum energy of low frequency fields, obtained at this stage.
IS
REFERENCES
1. Shapiro, V. D. Zhumal Eksperimental'noy i Teoreticheskoy Fiziki, 33,
613, 1963.
2. Drummond, W. E. , Rosenbluth, M. N. Phys. Fl. , 5, 1507, 1962.
171
3. Shapiro, V. D. , Shevchenko, V. I. Zhurnal Eksperimental'noy i
Teoreticheskoy Fizlki, 42, 1515, 1962.
4. Tsytovich, V. N. , Shapiro, V. D. Yademyy Sintez, 5, 1, 1965.
NONLINEAR PHENOMENA IN A PLASMA WAVE GUIDE
(ION CYCLOTRON RESONANCE AT A DIFFERENCE
FREQUENCY)
B . I . Ivanov
Ion cyclotron resonance (ICR) has been extensively studied both /178
theoretically and experimentally [see, for example, the summary in
(Ref. 1)]. Several works have appeared recently (Ref. 2-4) which
examined the problems of the nonlinear theory of ion and electron cyclo-
tron resonance. As is known, the non-linearity criterion has the
following form (Ref. 5, 6)
eEpl /. __^\ — ' ,
V$
(Eq — strength of the wave field; X — wave length; 3^ — retardation;
vq — ordered plasma velocity) . Formation of nonlinear phenomena is
facilitated during resonance (Ref. 6) , since in this case the non-linearity
parameter contains the additional factor "^ w (oj-to )~^. Thus, it is
possible that nonlinear phenomena may occur in the case of ICR, because
for this case the occurrence of large strengths of the wave field, small
phase velocities, and large wavelengths is characteristic. In this case,
nonlinear phenomena may play a significant role during heating (nonlinear
damping, nonlinear shift of the resonance frequency) and during the intro-
duction of high frequency energy into the plasma (interaction of frequencies) .
In principle, it is possible to introduce large UHF power into the
plasma at two frequencies , and then to perform ICR at the difference fre-
quency. Such a mechanism is also possible during the excitation of low
frequency oscillations in the plasma-bundle system (Ref. 7). As is known,
in unstable plasmas, low frequency oscillations, whose origin is
172
sometimes difficult to establish, occur simultaneously with high fre-
quency oscillations. The occurrence of the high frequency oscillations
is satisfactorily explained by the theory of plasma-bundle interaction.
One of the mechanisms leading to the formation of low frequency
oscillations may be the nonlinear interaction of high frequency oscilla-
tions [separation of difference frequencies, decay instabilities (Ref. 8,
9)] with the subsequent transfer of energy from high frequency oscilla-
tions to low frequency oscillations [parametric amplification (Ref. 10)].
This article makes an attempt to provide a model for this mechanism by /179
which low frequency oscillations are excited close to the ion cyclotron
frequency. The parameters of the apparatus are the same as in the pre-
ceding article (Ref. 12), which investigated the nonlinear distortions
of the signal form and the formation of combined frequencies . We employed
generators having a small power ('\'l w) which had a relatively small dis-
turbing influence on the plasma which was produced independently. In
order to fulfill the non-linearity condition, it was necessary to operate
at low frequencies (f '\' 1 Mc) and with low phase velocities (3$ '^ 10~^) ,
which, in its turn, made it necessary to employ a low-density plasma
(n 0- 10^ cm-3) (Ref. 12). On the other hand, f_ ^ f^^ >> v^^ (\)^^ "- 10^ p
— collision frequency) represents the necessary condition for observing
the ICR at the difference frequency. In view of these considerations,
the main ("beat") frequencies and the difference frequency were of
order of magnitude one: f i '^ f2 '^ f_ '^ 1 Mc.
In order to observe the weak ICR signal, a sensitive system of a
balanced, high frequency bridge was employed (measures were taken to re-
duce the noise level) . Figure 1 shows the diagram of the apparatus (1 — /180
current regulator; 2 — voltage regulator; 3 - high frequency generators;
4 — amplifiers; 5 — phase inverter; 6 — phase rotators; 7 — AVC unit;
8 — two-ray oscillograph; 9 — heterodyne receiver; 10 — heterodyne fre-
quency meter; 11 — self-excited oscillator; 12 — main anode; 13 — quartz
tube; 14 — water cone; 15 — auxilliary anode; 16 — cathode; 17 — mag-
netic field recorder; 18 — palladium filters). The plasma wave guide
consisted of the following, which were distributed coaxially: A plasma
core with a diameter of 1 cm, a lead tube with a diameter of 3 cm, a
copper casing with a diameter of 23 cm with the cross section along the
generatrix, and a solenoid. The total length was about 180 cm. The
quartz discharge tube was evacuated from both sides to a vacuum of
'^ 10~ n/m^, after which hydrogen was introduced from both sides through
the palladitm filters. When the entire length of the tube was continuously
evacuated, it was possible to obtain a constant pressure. Before the
measurements, the tube was treated to preliminary processing with pro-
longed, high frequency discharge (wavelength X = 2 m, generator power
P '^ 500 w) . The plasma was produced by discharge at a constant current,
and the discharge current was stabilized. In order to increase the
173
to pump
GDHZD
3 [^
10
ff
to pump
Figure 1
jnization coefficient, a cathode was employed made of lanthanimi hexa-
Dride with indirect direct current heating . Two anodes were em-
Loyed in order to obtain a stable discharge: The main anode and the
iixilliary anode. The anode potentials were selected according to the
Lasma noise minimimi.
The main frequencies fi and f2 from the generators were excited in
ae plasma wave guide by short spirals located close to the left end of
tie wave guide. An adiabatlc, absorbing water charge (length of about
cm) was located at the right end of the wave guide. The reflection
oefficient from the right end of the wave guide k equalled 0.1 - 0.3
Ref. 13). Thus, moving waves with main (fi and £2) and combined
nfi +mf2; n and m — whole numbers) frequencies could be propagated
a the wave guide. The output signal was employed on two spirals, one
f which was located in the magnetic field section which could be
odulated by a commercial frequency. From the receiving spirals, the
ignal was supplied to the two arms of the high frequency bridge. Each
rm consisted of an amplifier, a phase rotator, and the AVC unit. The
atter was used to eliminate relatively slow unbalancing of the bridge
""^ Avr ^^ ''■mod^ ' "^^ signals from the arm of the bridge were supplied
ut of phase to the heterodyne receiver adjusted to the resonance fre-
uency, and after rectification through the low-frequency filter they
ere supplied to the oscillograph. In the normal position, both arms
ere almost completely balanced, but the amplitude of the signal with a
odulating frequency remained somewhat larger. During the modulation
/181
174
Figure 2
of the magnetic field, at the moment the resonance value was crossed
(f_ — f„j) — twice during the modulation period — a signal for the
bridge unbalance was produced, which could be observed on the oscillo-
graph. This recording system has good sensitivity and noiseproof qualities
(subtraction of the signals from the two receiving channels leads to an in-
crease in the modulation depth of the carrier frequency by the ICR signal,
and simultaneously eliminates the plasma noise correlations along the wave
guide length) ,
Figure 2 presents oscillograms showing the dependence of the reso-
nance position on the magnetic field strength (fj^ = 1.5 Mc, f2 = 2.5 Mc,
f_ = 1.0 Mc; n = I-IO^ cm-^, p = 7-10-2 ^/^^ ^ ^a = ^ ka/m, H 1 56-62 ka/m) .
The upper line corresponds to the resonance signal which is inverted
during rectification. The lower line corresponds to the signal coming
from the generator recorder of the magnetic field. With an increase in /182
the constant magnetic field strength (for a fixed difference frequency
and a constant amplitude of the variable magnetic field) , the resonances
converge, since the resonance condition (f_ 1l Jci^ -^^ fulfilled in the
negative half period of the variable magnetic field.
Figure 3 shows the dependence of the resonance position on plasma
'\j
density (fi = 1.5 Mc, fa = 2.5 Mc, f_ = 1.0 Mc, H = 62 ka/m, H^^ = 7 ka/:
m.
175
Figure 3
p = 7-10-^ n/m% n :^ (1 - 2) 10^ cm
density, the resonances diverge, i.e
large magnetic field strengths.
^) . With an increase in the plasma
, they shift into the region of
In order to obtain quantitative estimates, the reflection coeffi-
cients from the wave guide ends, the magnetic field strength, the phase
velocity, and the plasma density were measured.
Dynamic measurements of the magnetic field strength were performed
by the generator recorder (Ref. 11) (see Figure 1). The self-excited
oscillator circuit was located in the modulated section of the solenoid.
Carbonyl iron was used as the induction core. Due to the small dimen-
sions and the small value of y ('^ 10) , the recorder disturbed the magnetic
field to an insignificant extent. With a change in the magnetic field /183
strength, due to the dependence y(H) the circuit inductance and the self-
excited oscillator frequency changed. The latter was measured by the
heterod3me frequency meter according to the zero beats , which could be
recorded simultaneously with the ICR signal by the two-ray oscillograph
(see Figure 2) . This system was calibrated initially by nuclear magnetic
resonance.
The phase velocity was measured by the system shown in Figure 4
(1 — Ave unit; 2 — phase meter; 3 — amplifier; 4 — delay line;
176
^^^^m^^i^^^^^^^^
.,^_
fTT=>r-
T-K^f--
r-^
^
^^
^
--
! — -\^^ — -jj^sy^-^" — — '5^::=ii;r7-"
K6?s5«w9©g9«^5^»oo<<)<»<^>^^
f
2
3
^
4
5
—
1
. L
/
J
1
J
Figure 4
5 — limiting attenuator; 6 — generator; 7 — selective microvoltmeter) .
A wave with the frequency f = f_ 2l f d was excited In the plasma wave
guide by the generator. A high frequency signal was applied to the two
antennae (a non-mobile short spiral and a mobile whip), and was supplied
to the phase meter input. The compensation method was employed to
measure 3$. A coaxial delay line, consisting of segments of the high
frequency cable (length, 2.5; 5; 10 and 20 m) , was switched into the
circuit of the mobile collapsible-whip antenna. These high frequency
cable segments could be subsequently combined in any combinations. With
a minimum distance between the antennae, the delay line was completely
introduced, and the phase meter indicator pointed to zero. As the dis-
tance between the antennae increased, the delay line decreased so as to
compensate for the phase shift produced. In spite of the fact that the
phase meter system was designed so that the phase reading was not depen-
dent on the signal amplitude, there was a possibility of error for small
phase shifts and significant changes in the signal amplitude. In order
to eliminate this possibility, an AVC unit was switched into the circuit
of both antennae, and also a selective microvoltmeter (for controlling
the high frequency signal amplitude) and a limiting attenuator (for main-
taining the amplitude at a definite level) were introduced into the cir-
cuit of the mobile whip.
/184
The length of the delay line AL changed linearly as a function of
the distance between the antennae £. The retardation was determined
according to the following relationship
ALV
=- ~3. 10-2
177
(e — dielectric constant of cable insulation) .
The electron density n was measured by the shift in the eigen
resonator frequency Af , and the dependence Af (n) was determined by pre-
liminary calibration with respect to the electron bundle (Ref. 12). The
relative content of atomic and molecular hydrogen ions in the plasma was
not measured.
The following conclusions may be drawn on the basis of the measure-
ments performed. At the moment that resonance is passed, the amplitude
of the "difference" wave increased. The resonance frequency increased
with an increase in the magnetic field strength, and decreased with an
increase in the plasma density, while f_ — fci« The picture observed
corresponds qualitatively to the excitation of ion cyclotron waves at
the difference frequency.
The experimental data agree qualitatively with the dispersion rela-
tionship for ion cyclotron waves
--(■-P'J)-
The quantitative divergences (stronger experimental dependence of resonance
frequency on plasma density) do not as yet yield to a satisfactory explana-
tion.
REFERENCES
1. Hooke, W. M. , Rothman, M. A. Nucl. Fusion, 4, 33, 1964.
2. Shapiro, V. D. In the Book: Plasma Physics and Problems of Controlled
Thermonuclear Synthesis (Fizika plazmy i problemy upravlyayemogo
termoyadernogo sinteza) , 1. Izdatel'stvo AN USSR, Kiev, 62, 1962.
3. Kondratenko, A. N. In the Book: Plasma Physics and Problems of Con- /185
trolled Thermonuclear Synthesis (Fizika plazmy i problemy upravlyaye-
mogo termoyadernogo sinteza), 3. Izdatel'stvo AN USSR, Kiev, 91,
1963.
4. Kondratenko, A. N. Atomnaya Energiya, 16, 399, 1964.
5. Faynberg, Ya. B. Atomnaya Energiya, 6, 431, 447, 1959.
6. Faynberg, Ya. B, In the Book: Plasma Physics and Problems of Con-
trolled Thermonuclear Synthesis (Fizika plazmy i problemy
178
upravlyayemogo termoyadernogo slnteza), 1. Izdatel'stvo AN USSR,
Kiev, 20, 1962.
7. Kornilov, Ye. A., Kovpik, 0. F. , Faynberg, Ya. B., Kharchenko, I. F.
In the Book: Interaction of Charged Particle Bundles with a
Plasma (Vzaimodeystviye puchkov zaryazhennykh chastits s plazmoy) .
"Naukova Dumka", Kiev, 23, 1965.
8. Orayevskiy, V. N. , Sagdeyev, R. Z. Zhurnal Teoreticheskoy Fiziki,
32, 1291, 1962.
9. Kondratenko, A. N. Zhurnal Teoreticheskoy Fiziki, 33, 1397, 1963.
10. Kino, G. S., Ludovici, B. Proc. IV Intern. Conf . , Uppsala, 2, 762,
1960.
11. Mints, A. L. , Rubchinskiy, S. M. , Veysbeyn, M. M. , Vasil'yev, A. A.
Radiotekhnika i Elektronika, 1, 974, 1956.
12. Ivanov, B. I. In the Book: Plasma Physics and Problems of Controlled
Thermonuclear Synthesis (Fizika plazmy i problemy upravlyayemogo
termoyadernogo sinteza) , 3. Izdatel'stvo AN SSSR, Kiev, 54, 1963.
179
SECTION IV
EXCITATION OF PLASMA OSCILLATIONS
RADIATION OF ELECTRONS IN THE PLASMA-MAGNETIC FIELD /186
BOUNDARY LAYER
V. V. Dolgopolov, V. I. Pakhomov, K. N. Stepanov
The cyclotron radiation of electrons in the plasma-magnetic field
boundary layer can make a significant contribution to the energy balance
of thermonuclear reactors with a small density, which employ magnetic
grids to contain the plasma. This problem was examined in (Ref. 1).
The thickness of the transitional layer between the plasma and the
magnetic field may comprise several Larmor electron radii
Pe = — [Ve = }/ -, tOfi = — 5 .
"*bV y tn' ° mc j
The trajectory curvature of electrons having a velocity on the order of
Vg is also on the order of p^. Moving along such a trajectory in a
vacuum, a non-relativistic electron radiates the following energy per /187
unit of time
S'-#^^ ^«-^ (1)
The number of emissive electrons per unit of layer area is npPg (n. —
plasma density). Therefore, the total intensity of cyclotron radiation
from unit of layer surface — if it is assumed that all the electrons in
the layer radiate the same way as in a vacuimi — is
3 (2)
, dw e^/Zo^flt/J
dt
«oP«-
Since
Bo — 8tz n^T, ^2)
^-""^V^' (4)
where a '\' 1. Expression (4) coincides, within an accuracy of a coefficient
on the order of unity, with the result derived by Burhardt (Ref. 1).
180
We may employ expression (1) only if the refractive index n for
radiated frequencies u '\' uig is close to vinity. In the case under con-
sideration, the radiated frequencies lie in the region of the anomalous
skin-effect
Re/z~Imn~^~f »1 (5)
(. = /:
ine^n,
A
— plasma frequency). In a medium with a large, complex
refractive index (5) the radiation intensity differs greatly from the
radiation intensity (1). For example, in a dense, non-relativistic
plasma I Q2^ — o)|) » the intensity of cyclotron radiation of the main fre-
quency decreases, as compared with radiation in a vacutmi, by a factor of
^ (Ref. 2-4).
Let us determine the intensity of cyclotron radiation, assuming that
the radiated waves correspond to stable plasma oscillations. In the case
of radiation equilibrium, the radiation flux falling on the plasma-magnetic
0) T
field boundary, Ipj = ,32 » equals the sum of the flux emanating from
the plasma I(w)and the flux reflected from the plasma IrjR (R — reflec-
tion coefficient). We thus find that I((jj)= Irj(1 - R) • In the case under /188
consideration, 1 - R '\^ — '\' — ^. The width of the radiation spectrtmi,
n c
caused by the Doppler effect during radiation and by the nonuniform! ty of
the magnetic field in the boundary layer, is on the order of a>g in the
case under consideration. Therefore, the total intensity of electron
cyclotron radiation in the layer is
/ ~ / ((b) COa .
i^%v^T e^n'gf'T^ (6)
A comparison of (6) and (4) shows that allowance for plasma polariza-
tion decreases the intensity of cyclotron radiation by a factor of
2
mc
Due to plasma resonance, radiation of boundary layer electrons strongly
increases in the region of frequencies o) which are less than the maximimi
Langmuir frequency fig* ^^^ considerably greater than ojg. In the resonance
region where Q (;t) = "l/ "^"^'"o W „,^ the intensity of electron braking
radiation greatly increases at the frequency o).
Let us investigate absorption of waves whose electric vector lies in
181
the plane of Incidence (the XY plane) , and who Impact on the plasma
layer from a vacuum. The electric field component which is parallel
to the plasma boundary, Ey "^ gikyy-iw ^ satisfies the following equa-
tion
^^ 7 P^'S- IT + i?^ — '^>'j^J'="' (7)
where e = l gJM ■ • ^ (x) Q' (x) ±s the dielectric constant of the plasma;
V (x) = - " f_MjO^ . — frequency of collisions; A — Coulomb logarithm (it
VmT^x)
c c
is assumed that v << to). Since Q. '^ — wb, the wavelength X = — is on the
order of the layer thickness.
At the point where Re e = (region of plasma resonance) , the wave
electric field increases sharply (£„ '^ E^ In e -^ Eq In — ; Ex '^^ — 2. "^ — Eq , /189
Eg — amplitude of incident wave) . This leads to the fact that a consider-
able portion of the incident wave energy is absorbed in the layer having
the thickness Ax ^ X—, in the vicinity of the resonance point. Therefore,
the intensity of braking radiation at the frequencies w < fi is
r r o ^ '""'o'^ (8)
c^m'^'
2
mc
Consequently, the intensity of cyclotron radiation is — ;j;— times less than
the braking radiation intensity.
2/3
We may employ equation (7) only in the case ofv>vo = B f2.
If V < vq, we must take into account the formation of plasma waves in the
resonance region.
REFERENCES
1. Burhardt, H. Nucl. Fus., 2, 1, 1962.
2. Ginzburg, V. L., Zheleznyakov, V. V. Izvestlya Vuzov. Radiofizika,
1, 2, 59, 1958.
3. Stepanov, K. N. , Pakhomov, V. I. Zhurnal Eksperimental'noy i
Teoreticheskoy Fiziki, 38, 1564, 1960.
182
4. Pakhomov, V. I., Stepanov, K. N. Zhurnal Teoreticheskoy Fizikl,
33, 43, 1963.
RADIATION OF LOW FREQUENCY WAVES BY IONS AND ELECTRONS
OF A NON-ISOTHERMIC MAGNETOACTIVE PLASMA
V. I. Pakhomov
As Is knovm, the propagation of three normal waves — Alfvdn, rapid
and slow magnetosound waves — is possible In a strongly non-lsothermlc
(Tg << T^) magnetoactive plasma, In the low frequency region (o) ^ ^^i) •
Each of these waves may be excited due to Ion cyclotron radiation or due
to Cherenkov radiation of Ions and electrons moving along a spiral In
this plasma.
This article determines the expressions for the Intensities of /190
radiation of these wave types by Ions and electrons. The emissive and
absorbant capacity of the plasma are determined In the frequency region
0) ^ '^x" ■^^ case of a low-pressure plasma is investigated in detail,
H
when the Alfvdn velocity v^ = yr rr considerably exceeds the speed of
sound in the plasma Vg i/-J^ • In this case, the refractive index of a
^M
slow magnetosound wave is considerably greater than the refractive indi-
ces of the two other waves. Therefore, it is natural to expect a sharp
increase in the radiation intensity of a slow magnetosound wave. It is
shown that in a low-pressure plasma the intensity of cyclotron radiation
. 2S+1
of a slow magnetosound wave by an ion for the s-th harmonic is
(^)
times greater than the radiation Intensity of Alfvdn waves and rapid
magnetosound waves .
Cherenkov radiation of low frequency waves by electrons of a non-
isothermic plasma may make a basic contribution to the over-all plasma
radiation. It is shown that the ratio of the intensities of ion cyclotron
radiation and Cherenkov electron radiation in the case of u) ffe ^"^1 ^^ °^
183
the order
-©"1"^)- -
ere Vrp- and v™, are the mean thermal
velocities, respectively, of ions and electrons, and v» is the phase
velocity.*
Propagation of Electromagnetic Waves in a
Non-Isothermic , MagnetoactiveP lasma
The general expressions which are given in the appendix in (Ref. 2)
may be employed to derive the dielectric constant tensor of a non-iso-
thermic plasma located in the outer magnetic field. The svmi with respect
to particle types — i.e., electrons and ions — must be taken. Let us
assume that the following condition is fulfilled
(Aa =
,— to
"j."tci
"Ha
<^\, a. =i, e;
2„ = rr^.
y-i k , v^
>1. / = 0, ±1;
■ h
^le = r^=
■fie
y2k,V.
»1, /=±1, ±2,
I "re
2oc = .
«1.
These conditions are fulfilled if it is asstmied that the plasma is greatly /191
non- is o thermic (T^ >> T^) . As a result, for the tensor of the plasma di-
electric constant we obtain the following expression
where
e<7 (k, w) = a8y -f chihj + dtijkhk +
+ e (x,- [xA]/ — r., [xA],) + / [xA]i [f.h]j.
u—\ '
c =
1 — u
+ v
n^ cos^0
q{Zoe) — l
d = - 1 ^ [j4r^ + ^ (20.) tg^ 9 ] sgn .
e^-iJS°'^
y^u cos'' Q
sgn (d; f — iy 2%
Mu cos 6 '
(1)
* A portion of the results have been published in (Ref. 1)
184
where
g {zoe) = 1+1 K'^zo.; P« = -f- = K ;;i?'' •
e is the angle between the direction of wave propagation k and the
direction of the outer magnetic field H. If the frequency u) is close
to the harmonics of the ion gyrofrequency stog^ (s = 2, 3, ...), the
following expressions must be added to (1)
a' =—c' =^id' =i=2ia„
where
n vs^ (s^n s i n G)^"-^ -2^
2's! pn I cos e I
(2)
0> SO)
Zs =
//«
K2 ^y^^ cos e
■^ c y Mel-
The dispersion equation determining the longitudinal refractive /192
index nn = n cos 6 as the function of the transverse refractive index
n_|_ = n sin 9 and the frequency w has the following form in the given
case [see also (Ref. 3)]
6 . 4 /" — 2 2 2 2u \ 2 r 9 "^1 + 2o , „
ni +n,(^—^n,-n,-^^n][nl-^^ + nUf~ (3)
~~M+ 2n!)] + ^ [nl {n! + n^) -n!n%] = iA,
where
+ u~\ + —Til— 1«. -- i7iri-/l + 2K' \n . [-V + V +
■ „2 [(H - T^)" + 2 2 9„2,«*l1 2/2, 2 A
185
When there is no damping (A = 0) , equation (3) has three solutions
1^11 = 1^ IP (J = 1» 2, 3). Assuming that
«! =n,i + inii, n\j<^n,j,
we obtain the following expression for ri\\^
A {"■ I /) (4)
The damping coefficient of the j-th normal wave is
"/^y^i/cosx. (5)
where x is the angle between the directions of the outer magnetic field
and the energy flux of the corresponding wave.
The dispersion equation (3) may be solved in two limiting cases : If
the Alfven velocity is considerably greater than the speed of sound in
the plasma (va >> v ) and if the frequency co is considerably less than /193
the ion gyrof requency (u << w^) .
Let us examine the case when v. >> v . The dispersion equation (3)
A S
assumes the following form
„« 4- A t-^f ni -ni)^n\ [n\ (^ + .;) - („
We may find one of the solutions for equation (6) by assuming that n| '^
nj_ "^ ng >> n^. As a result, we obtain (Ref. 4)
n n 1 = «s +
2 ,2 , "JL (7)
The imaginary part of the refractive index of a slow magnetosound wave (7)
xs
«!!
« I l"x
tW'-t^'')- <^>
The first component in the right part of equation (8) takes into account
Cherenkov absorption of a slow magnetosound wave in an electron gas. The
second component takes into account cyclotron absorption in an ion gas .
We may find two other solutions for equation (6) , corresponding to
186
Alfv^n waves and rapid magnetosound waves, assuming that nn '^' nj_ '^
'^ n^ « ng. The radiation of waves corresponding to these two solutions
was studied in (Ref. 5, 6).
In another limiting case, when the frequency o) is considerably less
than the ion gyrofrequency u^x (region of strictly magnetohydrodynamic
waves) , the dispersion equation (3) assumes the following form
n, —riA) [n, —n, [ris + rtA —n^LJ—riAnj. +
+ nl {tiA — ri]_)] =iA',
where
A' = -|/| • ^.[in', -n\) [nfin] ~n\ + nl) +
+ 2n\ hWa] + ^ [2n\ n\ [n] — n\—n\)— n\ {2n\ — n\ + n3i) ]) .
(9)
(10)
The solution of equation (9) /194
ni,=«^+o(-^) (11)
corresponds to an Alfvdn wave. The imaginary part of the refractive index
of an Alfv^n wave is
"'.-/:
-^n\(r^, + n\f (12)
n/n
Two other solutions (Ref. 7) of the dispersion equation (9)
n%.3 = \[nA+nl — n\ ±]/' [nl— n\^ n\Y + AnWr] (13)
determine the refractive indices of rapid waves and slow magnetosound waves,
The imaginary parts of the refractive indices (13) determining the damping
of magnetosound waves (Ref. 7) are
Az',2.3= y\
wn "s ("^1 2.3 - "^ + n\) + 2/t^, 2.3n^«i
187
General Expression f^ox A on Radiation Intensity in a
Non-Isothermic Plasma
Let us employ the general expressions (Ref. 4, 6) for determining
the strengths of the electric and magnetic fields produced by an ion
moving along a spiral in a strongly non- is o thermic magnetoactive plasma.
As a result, the components of the electromagnetic field produced by the
ion in the wave guide zone assume the following form
where
El,^s=PiSlnW,iE^J;
(i) .
H
£«=/'/ cos ^,y£l'];
//« = ±Pini cos W,iH%
(15)
/195
Pi =
2ea,;
"H^ V''
sin X
cos X •
d^n
rf
dn^.
"K/-«'OW/-"U)'
^s/ ={±k^sin')(. + ktiCosx)R—'o,jt + s^ + ^ — sf +
+ -^sgn(cosZ
£"?] = X («i£ii + n 1 1^33 — 611633) ^s — U-v («x — n i /«x —
L«x'o
• 633) '6i2 — V 1 eiife23 + Vinii-rij^ {isj^z — iHz ctg G)
Js-,
+
£■2 = ± COS X I y ^ [rti (/e^a — ^£23 ctg 6) — fsijEgaj/s +
7-(ny«i— n/633— "i^ii + 6iies3 + S23) +Uiirti//ZxX
X
X («/ — En) + u J 61262
/s| — sin X I f X [« 1 /"x (J6i2
^Kr,
IT (/J/^ 1 /1x — « 11 /«x6ll + 612523-) —
- £623 ctg 6) + 6112623] "^5 +
— V i (rt/rt^ii / — «/6n — n^i /611 + Si2e23 ctg e)j /jj ;
Hf^ = —Vi, cos 6 (f£i2633 + eiiiS23 tg 9) A —
— ~- [tijess — £11633 — £23 + £11623 tg 6) cos 9 + y B E^n • sin g] J
l-«x''o J
(16)
In expressions (16), v_|_ and vn are, respectively, the ion velocity
components which are perpendicular and parallel to the direction of the
188
outer magnetic field; rn = -— — Larmor radius of aa ion; the argument of the
<^i
Bessel function Jg and their derivatives J'g equals '^jj^o- The radiated
frequencies are
u>s/ = SMffj -i^k H (U>s;) V 5 .
(17)
In the case of s = (Cherenkov radiation) , equation (17) assumes the
following form.
(18)
In expressions (16) nj_ and kj_ are the positive solutions of the equa- /igg
tion for the saddle point
^'^»/(«x)_<^^tl/(^x) _,
= + tgx-
da,.
dk.
(19)
The negative solutions correspond to saddle points which do not lie on
the integration path, and therefore cannot make a contribution to the
integral. The upper sign in equation (19) , as well as in (16) , corres-
ponds to a cylindrically diverging wave; the lower sign corresponds to
a cylindrically converging wave. When there are several positive solu- -
tions for equation (19) , it is necessary to take the sum of all the solu-
tions. Each solution for equation (19) determines the relationship between
the angles x ^^'^ ^> i.e., the wave toward which the phase velocity is
directed at the angle 8, and the group velocity at the angle x to the
direction of the outer magnetic field.
We may find the intensity of cyclotron (s i= 0) and Cherenkov radia-
of an
expression
tion of an ion per unit of solid angle Wg.: by employing the following
where the bar designates averaging over time. As a result, we obtain
w^; = — a
sinX
'^f^APsj^''^^'''
cosX-
^'^■(^-^^^)hn«^.-«1.}V.;-4.)^
(20)
where
^./ = £^]5^] + cos(z + 8)fef.
189
In the case of s = 2, 3,
formula (20) is valid for the cases
of slow (vp Vrj,^) and rapid (v|| » Vrp^) ions. In order to determine the
radiation intensity at the main harmonics (s = 1), expression (20) may-
be only employed in the case of rapp.d ions. In the case of s = 1, the
results given in (Ref. 4) must be employed to determine the radiation
intensity of slow ions. For Cherenkov radiation (s = 0) , formula (20)
is also valid only in the case of rapid ions. Cherenkov radiation of
slow ions is greatly absorbed.
Expression (20) for ions with a velocity on the order of the mean
thermal velocity of plasma ions has the following form (in the case of
Ws,
where
Us:
\2s— 2
s-^{snJ^Y
«/«x«i;
2'=^(s!)2sinx
d%
cosx ■
11/
dn^
-'{Ai-<i)\Ai-'^\kf
X
X I [ + n? sin (x: + 6)(n/n j. — s^ sin 6 + isja sin 8 — h^s cos 6> + (s^^ —
— iHi) (S33 cos X qr U^.^ sin 1) + efg cos X — SggraZ cos X ] cos 8 [{s^i — ^
— i^ 12KH3 — ^623 ctg 6) — nfsss + efg] + cos (X. ^r 6) [(s^i — h^^) x
X (S33 — n^) — mn s / (633 cos 9 + is.^g sin e)]^}.
(21)
/197
Averaging expression (21) ovex the ion distribution, which is
assumed to be a Maxwell distribution, we inay determine the contribution
made by ion cyclotron radiation to' the emissive power of the plasma
at the 0) R; sa3jj£ frequencies
f}s}
(22)
(2iif''cii I / ?■
^i 92s— 2 '
where uq is the plasma density.
Cyclotron Radiation of a Slow Magnetosound Wave in a
Low-Pr4ssure Plasma
It was noted above that the dispersion equation (3) may be solved
if the magnetic pressure p-r, is considerably greater than the gasokinetic
pressure of electrons p^ = n^Tg (|:his is equivalent to the condition
^A ^^ ^s) • ^^^ ^^ investigate th^ cyclotron radiation of a slow
190
magnetos ound wave corresponding to the solution (7) of the dispersion
equation.
Employing the general expression (20) for the ion radiation inten-
sity and taking the condition n^ « n into account for the intensity
of the radiation of a slow magnetosound wave by an ion at the s-th har-
monics per unit of solid angle, we obtain the following expression
-2xjii?
(23)
where
U,i =
n\ sin .6
o^n^ J sin X
»-'^^(-'.^)
fn?sin(x — G)[r;xsin9x
X
X (rei2 — fs23 ctg 0) /s + (i^ sin 9 — a „ cos 6 j nf/,- o^ (%3f£i2 + '
£111623 tg 6) /s -f \~r S33 cos 6 -^ u', Btj_ sin 6 j nf/^ -f cos (X — 6) x
X [f X (eii4 + 633"^ 1) /; — (ZS12 — /S23 ctg 6) (^ tg 6 -f o , ) X
Kjirex/J I.
In the case under consideration, the refractive index is determined
by formula (7) , and the damping coefficient is
Xsl
(24)
7198
For ions whose velocity is on the order of the mean thermal velocity
of plasma ions, expression (23) may be simplified (in the case of s = 2,
3, ...):
»2,.,2r2
Wsl = ■
e'<xK,, _2.
2nc
Usie
(25)
where
U.
si
s2 (s^^_n^f^~^tl\n\ sin (X— 6)
22^(s!)?o(k— l)rt^,lSinX
cosX-
s 1
191
In this case, the equation for the jsaddle point (19) yields the following
dependence between the angles x ^^^ ^'
tge = («-i)tgx.
(26)
The contribution resulting frqm magneto-braking radiation of a slow
magnetosound wave by ions to the emissive power of the plasma, for fre-
quencies close to swjjj^, is
— ^Hi?o^(^P"x)^'~'"i "fsi"(fl — y-) -z"^
'" ~ (2jij'/=2's! vcn^n ^ , sin X | cos X| ■ ^ '"
I
In order of magnitude, ^s, --^ w^riA.,
2s— 2
(27)
~l , where wq. is the total /199
intensity of ion radiation in a vacutim. It was shown in (Ref . 6) that
the intensities of cyclotron radiatiion of an Alfven wave and of a rapid
magnetosound wave (j =2, 3) are
Ws2, 3 — teio«A
2s— 2
Comparing these results, we find that
"si
"52,3
■t;^\2s+l ^
Thus, in a greatly non-isothermic plasma having a low density, the
intensity of cyclotron radiation of a slow magnetosound wave for the
fVA2s-fl
s-th harmonics is, in order of magnitude, [~j times greater than the
radiation intensity of an Alfven wa;ve and a rapid magnetosound wave.
Cherenkov Radiation of Magnetohydrodynamic Waves by Ions
In the case of u « (jJjj^, the dispersion equation (3) has three solu-
tions corresponding to magnetohydrodynamic waves . The refractive indices
of these waves are on the order of jng (or n.), and are determined by ex-
pressions (11) , (12) . It follows from the condition of Cherenkov radia-
tion 3|] n||. =1 that frequencies oj << tog^ may exist during radiation
of rapid ions, whose velocity is
N
p/r
192
Let us investigate Cherenkov radiation of Ions in the low-frequency
region o) << okj- (u >> 1) . In order to determine the Cherenkov radiation
Intensity, let us employ the general expressions (20), assuming that
s = and u >> 1. In addition, let us assume that the arguments of the
Bessel function are small
k^ro = -p « 1.
(28)
Taking into account (28) for the Cherenkov radiation intensity of
an Alfvdn wave, we obtain the following
Woi =
8to,
u^ cos 6 sin X
n?sin3 6[4pi(«f-«^)+^?]
j[ _
2 d>
d^n ,1 , dn , ,
dn\
-e-2'oi«.
{n\~n\,f{n\-n\,)
(29)
/200
The damping coefficient of an Alfven wave is
"oi -if -Km
*°l=V^°^^K 8A?-
(30)
un\n\
Equation (19) for the saddle point assumes the following form in this case
n\ - — ' n\ tg X + n\ (nf - n\) = 0.
(31)
It follows from equation (30) that the Alfvdn wave is radiated within the
limits of a narrow cone ( X '^ ~ ) along the direction of the outer magnetic
field.
The Cherenkov radiation Intensity of rapid and slow magnetosound
waves per unit of solid angle, when condition (28) is fulfilled, is
i^02.3
^ ^°'"02.3 _ A 2.3 s,-n3 6 co^(Xj:J) \f^{n\ 2,3 -<)+2 ]^ _2,^^_
Stici
u cos^ sin X
cosX
d^n,
dn
2t3 D 2,3 1/2 2 \2
3«.
(32)
The damping coefficients of magnetosound waves are determined by
the expression
v.02.3 = -^ cosx K 87i? • ■
^02. 3 .
C
V\
"X2,3K2,3-«|3,2)
(33)
193
dn |u 2 2
The figure presents graphs of the fvmctions — — ^ and n||j(nj") for
the case nn > and n^ > ng. The points at which the curve "J inter-
dni
sects the line y = + tgx correspond to the solution of equation (19) . /201
As may be seen from the figure, a slow magnetosound wave (j = 2) repre-
sents a cylindrically converging wave. There are thus two waves for
each angle x *^ Xmax* ^^ ^^^ case of x > Xmax' ^ slow magnetosound wave
is not radiated. The angle x = X is determined by the following
equations
The rapid magnetosound wave (j = 3) represents a cylindrically
diverging wave. For any angle x> °^^ such wave is radiated. Expres-
sion (32) for a slow magnetosound wave is not valid in the limiting
case, when nj_ ->- o° and n?| -y n? + n^. In this case, we must employ the
general expression (20) for u >> 1 and s = 0. (We should point out that
Cherenkov radiation in a direction which is perpendicular to the direc-
tion of the outer magnetic field may be absent since the condition
B II n|| = 1 thus contradicts the assumption m « a^. .) If n. < Ug, then
the substitution n^ J iXq must be made in the figure. In the case of
n| < 0, the functions ^Hj change their sign.
dn^
Electron Cherenkov Radiation
It is of interest to study Cherenkov radiation of low frequency
waves (cj '^ '^±) by electrons of a non-isothermic plasma, because under
certain conditions it may make the main contribution to the over-all
plasma radiation. In particular, this is related to the fact that the
intensity of ion cyclotron radiation sharply decreases as one recedes
from the center of the line o) = saijj. (s = 1, 2, ...).
We obtained the expressions for the intensity of Cherenkov radia- /202
tion, by an electron moving in the plasma along a spiral, of each of
three normal waves. We determined the contribution made by Cherenkov
electron radiation to the emissive power of a non-isothermic plasma at
the (0 ^ ojjj-j^ frequencies.
We shall employ the general expressions given in (Ref. 2) for s =
in order to determine the components of an electromagnetic field
194
s •«
Figure 1
produced by an electron In the wave zone. As a result, we obtain
£,,=P,sin('F + a|-)£^.:
£x/=P;COs(w + a^)^^,;
//,/= ±P/n,cos(w + a^)^^,;,
Hxj = — tij cos (x + 6) £■?/.
where
a = sgn^cosx-^j.
£»,• == y J. (rtieii + n^, ,£33 — 611633) /o — y 1 [" I /ix ('612 — 1623 ctg 8) -h
£](/== ±COS^ (yx[«i(fei2 — 2e23ctg6) + ieiaSajj/o — t),[rt,/rex(n/ —
— i^ii) + Hi^23\ h] — sin X {ux [« II /"x ('612 — '?23 ctg 9) + ^iih^aVo +
+ , [tifnu - nUii (1 + cos== 6) + ef, + s^zj/oh
//^,- = — yiC0s8(tSi2e33 + Eii'S23 tg 9) ^0 +v,s\nt[eii{nj —
— ^ii) + 2i2e23 ctg 9 — efj] Jo-
(34)
(35)
In (34), V|| and vj_ are, respectively, the electron velocity components
which are parallel and perpendicular to the direction of the outer
195
magnetic field. The argument of the Bessel functions and their deriva-
vi
tives equals kj_ rg, where rg = ■ is the Larmor electron radius. The
= ZL
■^e
remaining notation was presented above.
Employing formulas (34) and (35) , we may obtain the following ex-
pression for the intensity of electron Cherenkov radiation per unit of
solid angle
nfiR'P^,
Wj =
SltMp,
\dn,
I]
[£;t/^W + cos(x+6)fe/)1.
(36)
da>
The argtmient of the Bessel function is small for electrons having
a velocity v_l which is on the order of the mean thermal velocity of
plasma electrons (vj_ '^ V'pg =
k±ro
e"):
m
«I.
In this case, expression (36) may be simplified
2x„ . R
Wi =
2m , \ d^n^j
0(1 — u) sin X cos X — 2~
dh
'If
d(o
{A-n\:f{n\,-n\,Y
(37)
where
/203
Averaging (37) over the electron distribution, which is assumed to
be a Maxwell distribution, we may find the contribution made by electron
Cherenkov radiation to the plasma emissive power at the frequencies
0) 'V av,. :
'^\
_ e==<flrto
";ig
(2't)'/'t;„
o(l — «)sinX
cosX ^^,^
{Ai—\^\-\i—\.Y
(38)
196
Comparing the contribution made. In order of magnitude, by ion cyclotron
radiation (22) and by electron Cherenkov radiation to the plasma emissive
power, we obtain
1, W; V*}'
(39)
It can be seen from (39) that the relative contribution of the elec- /204
tron Cherenkov radiation increases with an increase in the harmonic
number s, since It was assumed from the beginning that v^^ << v$ << v,j,g.
Let us investigate the Cherenkov radiation of a slow magnetosound
wave by electrons in a low-pressure plasma (ng >> n^;^) . We shall assume
that the argimient of the Bessel functions in (35) is small (kj_ rg << 1) .
This can be fulfilled only for very rapid electrons (vj_ >> v-pg) . Employing
the general expressions for the electromagnetic field components (34) ,
(35) , the equations (7) , (26) , and also the condition of electron Cheren-
kov radiation, we finally obtain
* 2«t<, • tg*X{sec'X-f,nlY ' p^ n^n^ | cosX| '
, Expression (40) determines the Intensity of Cherenkov radiation for
an electron having the velocity 3 1| per unit of solid angle in the direc-
tion X- If 3|| ng < 1, then the radiated wave represents a cyllndrically
converging wave (for x < — ) . The waves may thus be radiated in any direc-
2
tion X. The frequency of a wave radiated at the angle x is
(41)
y\sec'X~f,n^X\ ■
In another limiting case, when 3 || Ug > 1, the radiated wave repre-
sents a cyllndrically diverging wave. The vector of the group velocity
is thus directed at the angle x > Xmin' where
^min=^'"'=s^(P'"»)-
The phase velocity in this case is directed at the angle < 0_,g,.
in the direction of the outer magnetic field, while Qjjgj, = Xmi n • However,
we cannot employ the expressions obtained in the case of x "*■ Xmln' since
in this case m ->■ <", according to (41).
197
Averaging (40) over the electron distribution, we obtain the
contribution produced by Cherenkov radiation, by plasma electrons, of
a slow magnetosound wave to the plasma emissive power:
We should recall that at any angle x > Xmiji' electromagnetic waves with /205
two frequencies are radiated — one with the frequency o) > tOg- (electrons
with 3 11 Ug > 1) and the other with the frequency o) < uiyij^ (electrons with
the velocity V|| < v ). Expressions have been obtained (Ref. 6), in which
it must be assumed that Zq^ « 1, for the intensity of Cherenkov radiation
by electrons of an Alfv^n wave and a rapid magnetosound wave in a non-
isothermic plasma with a low density.
REFERENCES
1. Pakhomov, V. I. Zhurnal Teoreticheskoy Fiziki, 34, 16, 1964.
2. Pakhomov, V. I., Aleksin, V. F. , Stepanov, K. N. In the Book: Plasma
Physics and Problems of Controlled Thermonuclear Sjmthesis (Fizika
plazmy i problemy upravlyayemogo termoyadernogo sinteza) , 2. Izdatel'-
stvo AN USSR, Kiev, 40, 1963.
3. Kovner, M. S. Izvestiya Vuzov. Radiofizika, 4, 765, 1035, 1961.
4. Stepanov, K. N. Zhurnal Eksperlmental'noy i Teoreticheskoy Fiziki,
35, 1155, 1958.
5. Pistunovich, V. I., Shafranov, V. D. Yademyy Sintez, 1, 189, 1961.
6. Pakhomov, V. I., Stepanov, K. N. Zhurnal Eksperlmental'noy i
Teoreticheskoy Fiziki, 43, 2153, 1962.
7. Stepanov, K. N. Uspekhi Fiziki Zhidkosti, 4, 678, 1959.
198
BRAKING OF RELATIVISTIC PARTICLES IN LOW ATMOSPHERIC LAYERS
V. B. Krasovitskiy, V. I. Kurllko
As is known, V7hen a relativlstlc charged particle moves in an external
field, the braking force by radiation increases proportionally to the
square of the particle energy, and may be greater than the Lorentz force
(Ref . 1) . The influence of the medium on the braking radiation is of
great interest, since — when particles move in the lower atmospheric
layers — the braking force by radiation may differ significantly from
the vacuum force. The expression for the braking force by radiation of
an oscillator in a medium was obtained and studied in the non-relativis-
tic approximation in (Ref. 2, 3). The spectral radiation density was
studied in the relativistic case in (Ref. 4-7).
In order to investigate the influence of the mediimi on the braking
radiation, it is necessary to know the total energy losses of the particle,
i.e., it is necessary to specify the medium properties. We shall formulate
a model for a medium with an isotropic dielectric with a given dispersion /206
e(«,)=l ^
4iTe2no
where Q are the atom resonance frequencies; to^ = ; tlq — density
m
of the medium particles.
The following expression was obtained in (Ref. 4-7) for the spec-
tral density of particle radiation in the frequency region w >> Wtt (co„ =
'^ ^^l - B2 )
^("^ ^ -iTTf- '-?(' -pi){2/C|(^)- I^^(^)rf^j.P^e < 1;
Z' (">) = ^[l - -p^jjl + 2 iy_v. (^) - J.,, ($)] _ f [y_./, (^) _
(1)
whe
re £=-?.. Ji(i B^eV/i; j m and Kp(g) are the Bessel functions.
Let us employ formula (1) to determine the total losses by radia-
tion as a function of the particle energy. Since it is difficult to
199
integrate equation (1) in the general case, let us find the frequency
regions in which the spectral density has the maxima (K "^ 1) , and let
us determine the radiation intensity in each of these regions.
1. In the low-energy region, when the following inequality is ful-
filled
y < 5 « ;^ « 1. £ = (1 - p V^.,
(2)
the frequency u '^^ a)„ (1 - 3^) ^ — which makes the maximum contribution
m n
to the vacuum (Ref . 8, 9) — is small as compared with the resonance fre-
quency ojjjj << Q, In this case, the radiation spectral intensity has
maxima at the frequencies oij^ and Q. The braking radiation at the fre-
quencies ojjjj causes losses which coincide with vacuum losses
,2
"^B-^jr- £5- (3)
The maximum close to the frequency determined from condition g^e(a)) ^ 1
is also caused by the curvature of the particle trajectory. However, in /207
this case the influence of the medium is significant, as a result of which
the radiation is considerably greater than in the preceding case
W^p<{-Jfj'\ (4)
Thus, the braking radiation is larger (in the energy region under
consideration) than in a vacuum, and increases with the energy E propor-
tionally to E h.
2. The inequality (2) assimies the following form for given character-
istics of the medium and unchanged magnetic field strength with an increase
in the particle energy:
^"«J.«:-I«i. »)
In this case, the braking radiation maximtmi is located at the frequency
0) c^ ^ . 1 The corresponding loss is
"0
W^ — ."^"^* 1
f F- (6)
200
3. With an increase in the particle energy, the relationship
between the particle parameters and the meditnn assumes the following
form
ii,«#«^k<l. (7)
Thus , the radiation maximum in a vacuum occurs in the frequency region
m » Q. The force of the braking radiation depends on the parameter
y = . i in this case.
tojj E
In the case of vi >> 1, the braking radiation intensity in the fre-
quency region to 'X' oij^ is exponentially small as compared with the corres-
ponding value in a vacuum. The braking radiation intensity is also small
(U7^ «?£*")" ^^ *-^^ opposite case (y « 1), the influence of the medium
on the braking radiation may be disregarded, so that the particle losses
coincide with vacuum losses.
Thus , the medium influences the particle braking radiation only
when the inequality Wq >> ^a>jj is fulfilled. In the region of relatively
small energies /£^<^^] , the presence of the medium leads to an increase /208
r«l)
in the braking radiation, and in the energy region E^ >> — r- it leads to
a decrease in the braking radiation, as compared with a vacuum. Thus,
the braking radiation is significantly less than the vacuum radiation in
the energy region — > £ > ( — )
The change in the braking radiation intensity in a medium may be
explained in physical terms as follows. In the presence of the medium,
the frequency determining the maximum of the radiation spectral density,
according to formulas (1) , depends on the medium parameters
"'*"«^^^=7r^'<(^> = '
For small particle energies, the presence of the medium leads to an in-
crease in the frequency %(£ ('»%i) > 1» aiid the corresponding wavelength
V7
decreases): w < tOjjj < fi. Therefore, the braking radiation
201
intensity, which is proportional to ((») ) '^j also increases. For large
particle energies (01^ » U) , the frequency uij^ decreases considerably:
oj* « J2 << a)jjj (the characteristic wavelength X^ increases, e(a) ) < 1).
Correspondingly, the braking radiation intensity decreases. The
Cherenkov radiation, which can be computed from the customary formulas
in the case under consideration, is larger than the braking radiation
throughout the entire energy region where the influence of the medium
must be taken into account.
REFERENCES
1. Pomeranchuk, J. Journ. Phys., 1, 65, 1940.
2. Ginzburg, V. L. , Eydman, V. Ya. Zhumal Eksperimental'noy i
Teoreticheskoy Fiziki, 36, 1827, 1959.
3. Ginzburg, V. L. Uspekhi Fizicheskikh Nauk, 69, 537, 1959.
4. Tsytovich, V. N. Vestnik Moskovskogo Gosudarstvennogo Universiteta,
Seriya Fiziko-Matematicheskikh i Yestestvennykh Nauk, 11, 27,
1951.
5. Sitenko, A. G. Author's Abstract of Candidate's Dissertation (Avtore-
ferat Kand. Diss.), Khar'kov, 1951.
6. Kaganov, M. I. Uchenyye Zapiski Khar ' kovskogo Gosudarstvennogo
Universiteta, 49, Trudy Fizicheskogo Otdeleniya Fiziko-Matemati-
cheskogo Fakul'teta, 4, 67, 1953.
7. Vaynshteyn, L. A. Radiotekhnika i Elektronika, 8, 1698, 1963.
8. Schwinger, J. Phys. Rev., 75, 1912, 1949.
9. Landau, L. D. and Lifshits, Ye. M. Field Theory (Teoriya polya) .
Gosudarstvennoye Izdatel'stvo Fizicheskoy i Matematicheskoy
Literatury (GIFML) , Moscow, 1960.
202
EXCITATION OF WAVES IN A CONFINED PLASMA
BY MODULATED CURRENTS
A. N. Kondratenko
The excitation of waves in a plasma by modulated currents has been /209
studied repeatedly (Ref. 1-3). In all of these studies, it was
assimied that the plasma was an unconfined, isothermic, and linear
medium. In this article we shall study the excitation of waves in a
confined plasma, and we shall attempt to take its nonlinearity into
account.
When solving the problem of wave excitation in a plasma by a
modulated current j , we assume that the current is given, and we shall
disregard the inverse Influence of the wave on the motion of current
particles. This is valid as long as the energy losses of the current
particles at the wave length are negligibly small as compared with the
energy itself. The electric field strength of the wave is determined
by the equation LE = j , where L is the differential operator, which is
generally speaking nonlinear. A solution of this equation leads to the
following value for the Fourier field components: E = J. If the f re-
's"
quency of the current modulation is such that A vanishes (dispersion
equation), then the field strength becomes large. It Is usually assumed
that A is limited by particle collisions or Cherenkov absorption of wave
energy by plasma particles. This is the result of the linear theory,
and It is valid for insignificant amplitudes of the wave field strength.
However, in resonance, when A 'v. Q, the field strength greatly increases,
and the linear theory may be invalid. Even in the case of slight non-
linearity, the limitation on the amplitude of the wave field strength
due to nonlinear interaction may be more substantial than the limitation
imposed by dissipation.
Let us study the plasma layer which is infinite in two directions
(y, z) and bounded by two parallel metallic plates in a third direction.
The distance between these plates is 2a. A modulated current having
the form of an Infinite layer of thickness 2b, b < a, moves in the plasma
along the z axis. The plasma is located in a constant magnetic field
directed along the current, which is so strong that the plasma particle
motion across the field may be disregarded.
Let us Investigate two problems. In the first problem, let us
determine the wave field strength in the linear approximation for an
arbitrary dispersion law. In the second problem, let us determine the /210
wave field strength in the nonlinear, but hydrodynamlc approximation.
We shall asstmie that the nonlinearity is slight. The following quantity
is a small parameter of the problem
203
where e^^ and m^ are, . respectively, the charge and mass of particles of
type a (a = i — ions, a = e — electrons); Eq — amplitude of wave
field strength in the linear approximation; o) — modulation frequency;
v^ — phase velocity of a propagated wave, equalling the velocity of
current particles; u,,, = '^" ^^Ta — mean thermal velocity of a particles)
We can write the system of equations describing wave excitation in
a plasma under the conditions being considered as follows
x[(3-<)| • T + {3-2".\)|4"] = -4-|: (1)
1 _ gZ / V.
where E - z = component of the wave electric field; a = Ift = *
ightj ;
B2c2
4Tre2nr
(•
c — speed of light); ? = ojt - ksz; Q^ = — (no — equilibrium
density of plasma particles which is equal for ions and electrons) ;
u)' = 0) + ±\)^ (v^ — frequency of collisions between a particles) ; f Qct —
equilibrium distribution function of a particles noirmalized to unity;
Uj^ = _2L (V(jj — hydrodynamic velocity of a particles) .
The derivation of equations (1) and (2) is given in (Ref. 4). In
these equations, we must set y ~ 1» ^nd we must supplement them with
the boundary conditions , which in this case may be reduced to setting
the field strength on the wave guide walls equal to zero:
E{±a)= 0.
(3)
We can present the modulated current in the following form /211
'ep,V„ 0<{x)<b.
/=;iWsin?, /! = ( 0. b<ix)<a. ^""^
204
where e — charge; po — density; Vq — velocity of current particles.
We can obtain the equation of the linear approximation for the
field from equation (1), setting u^^ = in it. The solution of the
linear equation can be written as E = R(x) cos 5. We obtain the
following equation for R(x)
g-l-A^2/? = -47««aA. (5)
(5a)
(6)
where
;fe2 =-o(o''l 1+5]
\ «=,
Solving equations (5) and (3) together, we find
R(x)= ^/?„cosa„jc,
n=0
where
Jin = -^ — ,ln — ^ -y-^ sm a„o,
n 1
a„ = ^(2n+l); A„ = a^-&i.
If the velocity of the current particles is close to the mean
thermal velocity of plasma ions or electrons, the integral in formula
2 2
(5a) must be determined numerically, Im kj_ and Re kj_ are equal in order
of magnitude, damping is large, amplitude of the excited wave field
strength is small, and the energy losses by the current are insignificant.
In this case, we can confine ourselves to the linear theoiry.
2 2
If Vq >> v-fg ^ or v-fg >> Vq >> V.J1J, then Im kj_ << Re kj^ and at a
certain frequency of current modulation | A^^j "^ Iva. k_j_ << o^^ may hold.
The corresponding K^ (we shall designate it by Eq) increases greatly,
and only the resonance component can remain in the sum (6) : R(x) ^
!^ Eg cos Oj^X.
Since the amplitude of the wave field strength is large, the non-
linearity of the medium may be significant. When determining the non-
linear dispersion equation, let us regard the nonlinearity and thermal
scatter as small independent additions to the linear hydrodjmamic
205
2 2
dispersion equation (A^ = 0, a^^ = Re kj_) . Therefore, the wave damping /212
caused by kinetic phenomena can only be taken into consideration in
the third approximation.
We may find the hydrodynamic velocity of the linear approximation,
corresponding to A^^ I^ 0, from equation (2)
«(') = Sa cos oLnX sin ?. (7)
Let us substitute (7) in equation (1), and let us retain the terms "Xiz .
Then, representing the field E in the second approximation in the form
E^ ' = EgRaCx) sin 2^, we obtain the following equation for R2(x)
-^ + q^R% = h cos^ a^x.
(8)
a=l
Solving equations (8) and (3) together, we find
i?3=A^_/^._J^.£2iif_^.^2!!f^ (9)
In the second approximation, the hydrodynamic velocity is
ui'^ = ~^^R.
,2
C0S2S. (10)
After substituting u^-*^), u(^) in the right part of equation (1) of
a C(
the first and second approximation, we obtain the following equation for
the field E in the third approximation
-^ + --' [y - ^'^> s , ;r .., r - -43 cos 3^ -
<-(--+ '\)(l-"x\)
' Zj tt; 9T2 (-^ — «™ I ^a + -o- — ^ — -^5 —
(11)
X cos^ a„x ] cos a^x cos ?,
where Aa is a certain function of x which is determined by the third
206
harmonic with respect to K'
We can write the solution of equation (11) in the form E^^^ = /213
= EgRaCx) cos 5. We obtain the following equation for RsCx)
"S?' + *i ^3 = Qi cos a„x + Q2 cos gx cos anX + Q3 cos 3a„x, (12)
where
-4. y
-V . 3
n
9- 8ui
qi ' 8 ^" ^
^4(i-u2J2 9"^ W4a2_92 COS 9a •
Since Qi 2 3 '^ e^, we can write the solution of equation (12) in
the following form (Ref. 5)
/?3 = cos']> + BiCos(a„+ q)x + B2Cos{o.„— 9) a: + B3 cos 3a„A;,
a=^.+sM.
We thus find
cos
(b -.^\y^Q2 p°sK+g)x cos (a„ - g) X -\
-^ cos 3a„jf.
Employing the boundary condition (3) , we find that
(O, ^ n 2a„sin(7a x_ „v
^«xy g ^a^ — cl"
We can write the solution of equation (13) as follows
We find „ ^ ^ 4a^Q
2a„8 = Q,+ j:p^,smga
(14)
207
Thus, we can write the nonlinear hydrodynamic dispersion equation I2\.k
as follows
i-'+? .(.+<v.K.- ^]°'.+^-'- (15)
and V includes both pair collisions and Cherenkov wave absorption by
plasma particles.
In the case of resonance, when Re k_|_ = a^, the field strength ampli-
tude of the excited wave is
^8 —
|(^-)-{"ET^.-5/f
(16)
Let us first examine high frequency oscillations: Up. g << !• Dis-
regarding the electron mass as compared with the ion mass , we obtain
2a„8 = -as^Qg 8,; 8, = | . . ^""'"^^ - ("°--"')'' . !M£ (17)
' 0.1. X 8 4^._22^ Qg.C-'-Sy "" *
We can show that in the case of (u ^ ^Oe> 5i > always holds. We obtain
the following value from the equation (15) for the square of the phase
velocity
P Qg, - <o» + c^.l L^"' " + Sg^ - .^ + ^^l J • (18)
Since 6i > 0, the phase velocity increases with an increase in the field
strength amplitude of the wave. We may use formula (16) to determine
the amplitude of the field strength:
£. ^ S'toiepoV, sin a„ft
" ^i{(,xr-v\i]Y ""'' • <i9)
9 *—
If e 6i > — , the maximum amplitude is determined by the nonlinearity ;
8^,Vt ^gp. sina„ft (20)
<8
"1
208
The low frequency oscillations: iijig >> 1 >> "xi*
The amplitude of the wave field strength in this case is /215
'" ^.[c*)-+e)"r °"° •
HS)
If «?eF^> 1 ((* = — ), then 62 and the phase velocity have the values
of (17) and (18), respectively, for a replacement of the indices e ->- i.
q2
In the case of sound oscillations, when ">*^— ^, we have
(22)
Vl = l^v
^li (1 + ^'»2) (23)
" ^l + ->'r
If — ^ < e?6„, the field strength amplitude is limited by the nonllnearity
and has the same form as in the case of high frequency oscillations (2) ,
with a replacement of the indices e ->• i, 1 -> 2.
REFERENCES
1. Kippenhanh, R. , Vries , H. Zs. Naturforsch, 15a, 506, 1960.
2. Kondratenko, A. N. In the Book: Plasma Physics and Problems of
Controlled Thermonuclear Synthesis (Fizika plazmy i problemy
upravlyayemogo termoyademogo sinteza) . Izdatel'stvo AN USSR,
Kiev, 176, 1963; Uspekhi Fiziki Zhidkosti, 7,371, 1962
3. Aleksin, V. F. , Stepanov, K. N. Zhurnal Teoreticheskoy Fiziki, 34,
1210, 1964.
4. Kondratenko, A. N. Zhurnal Teoreticheskoy Fiziki, 34, 606, 1964.
5. Bogolyubov, N. N. , Mitropol'skiy, Yu. A. Asymptotic Methods in
the Theory of Nonlinear Oscillations (Asimptoticheskiye metody v
teorii nelineynykh kolebaniy) . Fizmatgiz, Moscow, 1963.
Scientific Translation Service
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